I downloaded a building with extra useless space made as fake shops downstairs and I want to use that space and make a gym out of it since my town lacks a gym. Does it work if I make it as a public room with an entrance from the street? I have one more building which would make a nice spots bar in the ground floor, I hope it's possible.

Not really, because on a residential lot the Public Room Marker only applies to those who live there, appear to live there in the case of NPCs hibernating in the private areas behind NPC-only doors, and those whom the resident household has lot-greeted. Other sims in town will not have access and won't even recognize the space as usable to them unless the household living there invites them over or throws a party there.

You could try this the other way around, that is by designating the lot as community rather than residential and using NRaas MC > Add Sim to force a household to live on it. There are some downsides to this structure such as the household living there won't have a usable mailbox (but inactives don't tend to care) and if using SP Money's Unified Billing they will have to pay property taxes on probably the entire lot or at least more of it than is really meant to be their living space.

Maybe there is another fancy solution that I am missing.

Tedhi made some wonderful residential lots with shops on them, for example a dairy farm here: https://www.youtube.com/watch?v=fVWj6yPbjZE

She explains how to do it here: https://www.youtube.com/watch?v=bqPDngYTm1A. Not sure how well it works for non-active households though.

Do NPCs living in apartments ever actually use the public areas? I've never witnessed it before. In my game they just go to work (to tend bars etc) in the morning and come home pretty late at night. TBH they usually don't have that much time to hang out because I use the 23hr bars mod so they are constantly working.

Yes, they do, but I think there has to be something in the public area that's worth visiting. I built a few places that have a hot tub room and did notice NPCs taking a dip in the tub from time to time. Also, it's hard to catch them using a public area, because most of them will use it at odd hours of the morning when your sims are sleeping and the game goes into fast forward.

I have done this as well. I put in a pool, hot tub, some grills, picnic tables in the common area and I do get the NPCs that live there coming out to hang out. Also if you have register when you see those NPC's change their job. I set a couple to retired and boosted their money so they wouldn't need to work so they would hang out more. I also go around and knock on their doors to make sure I build a friend ship level and I will host a party so they all come out.