We have always tried to give our cats the space they need until they trust us enough - but that isn't always possible. We do worry about bites, because they can be serious - even little kitten bites. Scratches are far more likely than bites, though.
At 6 months they've most likely got their adult teeth in.
We used a trap when we needed to transport the outside cats to the Vet, and they always loaned us a carrier for us to return the cats.
For the two different batches of kittens, once we'd gotten to know them, we set food inside a crate to coax them in there - we never used our hands.
You may already know this, but many cats, especially ferals, get overstimulated easily. Watch the cat as you're petting it - when the tail starts to twitch or swish, even just a little bit - stop petting. Like I mentioned - they're far more likely to scratch than bite, but cat scratches run all the same risks as bites and must be treated right away.
We have one little kitty that never gave "the signal." After a good scratch, the way we handled him was to pet him a couple of times - then stop. If he leaned in for more, then we continued.
To wash the cats, once we brought them in - well - we live in an RV, so our only choices were the shower stall, a teeny bathroom sink - or the kitchen sink. We decided to use the shower. And believe me, biting was not the problem. Of the three, only one has SERIOUS problems with being bathed.
But I have serious allergies to cats, so that bath happens once a month. Gary does it fully clothed in thick clothing and just tries to watch his hands. The clothes give Shelly a way to "climb up" Gary - but it doesn't hurt nearly as much as it did when Gary just wore a bathing suit!