Give him the house for the night and YOU go to a hotel.

OP said "Let’s say the priest has “friends” (both male and female) and has known them for many years. The priest is obviously too familiar with his single female friends, but he spends the night at their homes, sleeping separately? Of course, some boys either brothers or cousins are sleeping in the same house, too."

So, yeah, if they can all sleep in one hotel room.  Otherwise it is more frugal to give the priest money for one hotel room.  

If there’s no motels nearby and only a widow old enough to be the priest’s grandmother?  

There are some priests who travel with a small pull behind camper.  

This is just as a matter of principle, not necessarily with regard to a particular situation. I don't consider myself a prude or scrupulous, but I think it's wrong to house a priest in a private home unless he is a relative (for example, a cousin, a nephew, if not a son or brother) and unless a married couple or widower also resides there.  This is mostly because of the possibility of scandal because of the unfortunate reputation(s) too many clergy have earned, in general.  The public is not entirely accurate on their judgment of priests, to say the least -- which obviously is not terribly fair to priests, but it is nevertheless a reality that has to be dealt with.

There is another, perhaps surprising, perception that can result; it did in a situation we had in our parish a couple of years ago.  Yes, the person on the previous page who said priests have to sleep somewhere when they lack local housing, such as a rectory, is correct, and this was very true in the case I cite as well.  The priest in question was unwelcome in a shared rectory because he belongs to a trad apostolate (but is not an independent) and the other priests do not.  To avoid the discomfort, the priest accepted an invitation to stay with a family who were parishioners.  First of all, given that there were several teenaged girls living there, I think the priest should not have accepted that particular invitation -- again, for perception reasons, even though I'm sure none of us doubt his fidelity to his vows.  Second, because the priest felt (naturally) grateful to those parishioners, they became obvious favorites at parish events and such. The favoritism was rather undeniable. Had they not been parishioners, this would not have been a problem.

This year our parish is once again in this situation, but with a different priest, since the other one was transferred.  Seriously, most of us parishioners just want his order to buy a darn piece of property already, since they can afford to.  (Basically, purchase an off-site rectory.)  They haven't yet; we keep hoping.  The other thing that's happening, housing-wise,in our region, is the return of "in-law" and other cottage-type units.  That's also a solution if either the priest's congregation or order can afford to subsidize the unit; however, an outright purchase would obviously be far preferable.