I had my air conditioner serviced. They sent a woman tech out. When the job was "done" and she departed, I discovered the woman left the indoor unit housing cover and screws off, laying on the basement floor; and outside, she left a metal plate from the outdoor unit laying in the grass.

Report her and get your money back, if possible. She’s either lazy and/or doesn’t know her business.  If I have to do half the job, I may as well do the entire job myself.  
An able-bodied woman with brain cells intact can learn HVAC.  Especially when it’s a woman doing a job considered “male,” it’s extra important her work be of excellent quality. 
Before blaming the shoddy job on her sex, do you know her situation?  Is she in need of a job that pays a living wage?  HVAC is an area of need that doesn’t require a degree, can be learned on the job, and has promotion potential for a decent wage. 
Maid at a motel requires about the same physical agility and stamina, but a single woman working 60-72 hours per week won’t make half as much as HVAC. In many areas, a motel maid will be unable to support herself.  
Living wage jobs for traditional Catholics are getting increasingly difficult to find. Among that population, jobs for single women who must support themselves are even more difficult to find. 
Nonetheless, there’s no excuse for shoddy, incomplete work. Just about anyone can reattach covers with screws and nuts!  

Good points, Matthew and Seraphina, but I decided not to report her. I figure maybe she was tired and had some brain fog which led to her absent-mindedness. I can understand one of the incidents happening but BOTH happening at the same time in two different places (basement and outside).  Good thing I saw the plate before running over it with my lawnmower. Anyway, the AC is running great. I'm sure if this is a habitual problem with her, she's already been reported or she will be in the future.

Hi Croix

Anyway the biggest question here is that there's female HVAC techs? Those kinds of trades (anything labor-intensive like that, automotives, HVAC, culinary industry) have historically been very male-dominated.

This isn’t Croix!
Yes, historically, HVAC is a male job, but really, the lady of the house is more concerned with the air temperature within. Among the Amish, very traditional in male/female roles, the woman regulates the cook stove and the man, the heating stove. 
There’s nothing inherently male or female with servicing an AC unit. 

Good points, Matthew and Seraphina, but I decided not to report her. I figure maybe she was tired and had some brain fog which led to her absent-mindedness. I can understand one of the incidents happening but BOTH happening at the same time in two different places (basement and outside). Good thing I saw the plate before running over it with my lawnmower. Anyway, the AC is running great. I'm sure if this is a habitual problem with her, she's already been reported or she will be in the future.

You should report her so that she gets constructive feedback. Attention to detail is key in any technical task, doesn't matter if it is sewing, cooking, or working on hvac. After the work is done, it is part of any SOP to walk the area to make sure they left it in the same condition as they found it. The next time the situation could be worse due an expensive tool left behind or a safety issue that resulted a death. A good employer would address incident not necessarily the person. There could be possible gap in training or expectations from the employer. If there is a pattern then it would require disciplinary action. When you report, it is a good practice to speak of good as well as the bad just to show that the experience was not a complete failure.

A few years ago I had a new furnace installed. Three men did the work, ducting, steel gas line etc. Year later , the inspector failed the installation.
Wrong glue, leaking hoses, bad hangers, air returns falling apart!!
So,  the workmanship is getting sloppy...this coming generation is missing something...attention to detail.
Anyway the furnace guy got a fine, even after he tried twice to fix things! GO figure!??