Replacing cat-5e with cat-6 (or something else) for POE

About 8 years ago, I had my house wired with cat-5e (16 runs including one to a driveway gate opener/intercom - all the others are inside only). The house is 3 stories, about 75 feet wide. I went with WiFi security cameras because they were easy, so I don't have ethernet to the cameras (and doorbells). I don't think this is a project I would try to DIY. Has anyone done something like this, and what should I know before I try to hire someone to do this? How much did it cost you? Is cat-6 a good option these days for gigabit+ wired networking with POE? I read that cat-5e (especially over long runs) can be a fire hazard with POE - is that for real? Thanks for any thoughts/guidance!