1) Explorer...open explorer, press f12, which will open developers tools...select Cache off the menu bar and Clear Browser Cache.
2) With Firefox its in history....Clear recent history, click Everything...Then scroll down and select which elements to clear, and select all....Click "Clear Now"....Close it out then reopen.

Am I imagining things? Do unused icons on the desktop slow down a computer's speed?

There are people far more proficient than myself in computers, code, etc...but 50 icons are a lot to have on a screen.  If you saw a difference in performance, I would say it might be because if she doesn't have an actual graphics card, any "graphics" computing is being pull from the intergraded card, drawing on the processor needlessly.

1) Delete/Uninstall any programs that are not being used
    A)Get rid of all bloatware, free trials, junk ect.
2) Organize the remaining desktop into maybe 10 max icons/folders full of icons.  
3) Turn off visual effects, this my give performance a little boost. (control panel...performance and tools....Adjust Visual Effects...go for performance...done)

She's using Avast free antivirus and Malwarebytes (free version), but they don't seem to slow down the processor at all.

1) Is all her antivirus and malware definitions up to date? 
2) Have they been set to update and scan on a daily basis?
3) Are there any pending Windows updates/security fixes?
4) Just because she has the programs running in the background doesn't mean she is protected from malware or viruses.  She will always need to keep them up to date for the best chance at being protected.  Some attacks turn off the automatic updates and leave system venerable to future attacks.

I'm wondering if anyone here has any more suggestions.

Just a couple
1) Autorun...Hate this...Things running in the background needlessly...Type in MSCONFIG into the start bar and go to the Start Up tab.  Anything that is not Microsoft/antivirus/malware/hardware support can be disabled. This is one thing that should help out a lot.  Most programs demand attention upon boot and want to be high priority.  You don't need Java updater, Quicktime, a dozen other things that want to be running, hidden in the background. After disabling most of them.  Then reboot...the boot should finish faster.
2) I don't know about her computing habits, but its a good idea to reboot daily.  If anything errant is hung up or running in the background, it will at least close it out.
3) While 4 gig of ram may be adequate, its definitely not a whole lot.  My cell has more...If she has a slot free, an extra dimm shouldn't be all that much.  Just depends on age, mother board, etc...but there should a noticeable difference with a small upgrade....Its a lot cheaper than buying a new rig.

The single solution. This will work. 
Have her clone her current hard drive to an SSD.
The easiest way is to go to a computer shop so they can clone her drive.
No need to buy a new PC. In fact, her current, older PC with an SSD (solid state drive), will be faster than any new PC that is equipped with a standard mechanical hard drive.

The most economical way to do this is find out what is her current hard drive capacity. (I.E. 500GB Hard drive) She may only have only a fraction of the capacity used up. 
So let's say she has 100GB used out of her 500GB hard drive. She could buy through amazon a 250GB SSD (I recommend the Western digital blue SSD or a Samsung 850 evo SSD) and go to a PC shop and have the current mechanical hard drive cloned to the SSD drive with a program called Norton Ghost (or any other cloning software). For Windows 7 I like the simplicity of Norton Ghost 11.5.
Right after the cloning process, they unplug the old hard drive cables and just leave the new one active.

Voila! I guarantee she will thank you for her "new pc".


 

The single solution. This will work.
Have her clone her current hard drive to an SSD.
The easiest way is to go to a computer shop so they can clone her drive.
No need to buy a new PC. In fact, her current, older PC with an SSD (solid state drive), will be faster than any new PC that is equipped with a standard mechanical hard drive.

The most economical way to do this is find out what is her current hard drive capacity. (I.E. 500GB Hard drive) She may only have only a fraction of the capacity used up.
So let's say she has 100GB used out of her 500GB hard drive. She could buy through amazon a 250GB SSD (I recommend the Western digital blue SSD or a Samsung 850 evo SSD) and go to a PC shop and have the current mechanical hard drive cloned to the SSD drive with a program called Norton Ghost (or any other cloning software). For Windows 7 I like the simplicity of Norton Ghost 11.5.
Right after the cloning process, they unplug the old hard drive cables and just leave the new one active.

Voila! I guarantee she will thank you for her "new pc".

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It would seem you're presuming it's a desktop or tower computer. But it's a laptop.
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So for me to clone an SSD and leave the old hard drive cables unplugged, would the SSD need to be an external drive? 
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Not to "derp" things up, but does this person ever, you know, shut it down?

I suppose the basic point I'm making is, make sure your first checklist is the basic basics, such as "Is it plugged in?", "Is it turned on?" "Have you blown the 'Cheetos' out of it?" etc., etc. etc.

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Not a bad idea to re-think the basics! I think I've covered all those, but I can't be absolutely sure, since there's still a problem. So maybe I've overlooked something basic. This laptop spends most of its time SHUT DOWN. But come to think of it, I'll have to find out HOW she shuts it down. This is a Windows 7 OS, so you're supposed to click on the blue circle in the bottom left corner (says "Start" when you hover the mouse over it) and select Shut down. Then you have to wait while the operating system closes down normally which takes a minute or two. I knew a guy who is real impulsive like the owner of this computer is, and I had to watch him when he shut down his computer when I came over to help him when he was having problems with it. He had a tower HP desktop with an ON switch on top of the front face frame. I couldn't believe what I saw when he was turning off his computer he pressed the ON switch and held it down until the computer suddenly blinked off and the screen went dark. I asked him if he always turns it off that way, and he said, "Yeah." 
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I explained to him that could be most of his problem right there. He had no idea what I was talking about. I said that the computer has a list of things it needs to do in order to shut down normally and you give it the chance to do those things by clicking on the "Shut down" option that appears on the screen. He admitted that he didn't like to wait for that, PLUS he is afraid to leave the computer on when he isn't right there in front of it because he thinks it might catch fire. So when he has to go to the kitchen or answer the door, he usually presses the ON button and holds it down like that to be sure the computer is OFF. I explained to him that every time he forces it to crash like that he does a little damage to the hard drive, so he's wearing out his drive a lot faster, plus it could be causing the system to run slowly because the drive has to jump over the damaged places and re-arrange all its disc information. He said, "Oh, I didn't know that; I thought I was just turning it off like a desk lamp or a radio." 
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He drives a Mercedes Benz Diesel sedan, and when he starts the engine he NEVER waits for the glow plugs to warm up but sticks the key in the ignition switch and twists it letting the engine crank over until it starts. I told him he's wearing out his starter, his battery and his ignition switch like that. He didn't care. About two years later he had to spend $2,000 fixing the ignition switch and starter plus he wore out two batteries. The battery is HUGE, about 18 inches wide and costs about $180.
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You never know how someone uses a machine when nobody's watching.
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The laptop power is okay (battery fine, charger okay, plugged in, charged 100%, then unplug charger to avoid over-charging, don't let the battery go below 30% before charging again). Next step might be to open the cooling ports (remove screws on the bottom and remove the panel covering circuits and stuff) and remove any dust from the fan and heat sink area. I had a laptop that got clogged with cat hair in a corner pocket that wasn't obvious at first. When I used a pipe cleaner with a hook bend in the end I was able to drag out a clump of hair and that made the operating temperature improve a lot. But that wasn't this laptop.
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If this was an Apple laptop it would be a service call because they don't want owners to open up the notebooks. But I've known of guys who do it anyway. Basically it comes down to ruining the first computer in the process of learning how to open it up -- then you can use the experience to safely open the second computer.
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This would have been nice to know years ago.

Even though I now have a computer that is less than a year old (I purchased it last summer) I ran this msconfig--going through the search.  I was able to view the programs that run on starting up the computer and saw a couple of programs that I know I have no use for:  Microsoft One Note which, apparently, allows your personal docuмents to be accessed by others since you don't really have any control over them, and a Garmin program--how often do these people think I need to update my Garmin???!

I'll have to research the other programs--they might just be necessary programs for the ordinary use of the machine.

Desktop icons make no difference. Literally 0. 

If a newish computer is running slowly, the problem is almost certain to be either a virus or that there are garbage programs running in the background, which are basically the same thing. 

What I would suggest you do - bring up the task manager (ctrl+shift+esc) and go to the Processes tab. Select the "show processes from all users" option (near the bottom of the window), take a screenshot of that, and post it here. It tends to be immediately obvious if there are issues, if you know what to look for. 

Then go to Control Panel, Add/Remove Programs (or whatever that option is called these days), and post a screenshot of all installed programs here as well. New computers come with a ton of random software that serves no purpose installed, and cleaning all that up can help.