Stranded wire is fine. I'm guessing that he's not in a 15,000 sq ft mansion, so there's no need to worry about pulling cable to the 328 ft limit. And it sounds like he already has the cable... no point in going through the hassle to replace it.
I'd recommend finding a good place to homerun the cables to.... a media cabinet in the basement, the office, or some other place that makes sense. I wouldn't run everything to the garage. If the FIOS router needs to stay in the garage, then just run an uplink from the FIOS router to your new switch location. If you can move the FIOS router, that would be better.
I would also recommend getting a patch panel. You could say it's overkill, but like the recommendations above, it's better to terminate the wiring to fixed equipment and then run short jumper cables to the switch. 1) it will look much nicer. 2) your wiring will last much longer than that switch, so you don't want your cabling cut to a perfect length to match that switch.