Wow, you really seem to have hit the ground running! It hasn't been an easy year for gardeners so far, but you seem undaunted by the fickle weather.
A few attempts at answering your queries:
The herbs all behave a bit differently but most of them will come back after cutting. Mint and lemon balm are vigorous and will take over if you don't watch them (mint in particular is better in its own pot. But you can move some to a separate pot.) Rosemary will grow to a sprawling bush and will also need a space of its own eventually.
Chives will grow back after cutting, although you may have to wait a while, so it's best just to cut a little as you need it. Coriander is an annual and will die this year, but if you let it set seed you'll probably get new plants coming up next year, or you can save some seed and plant again. It doesn't really come back much after you cut it. All the rest will carry on growing after you cut them, and will over-winter with no problems. Mint and chives and lemon balm will mostly disappear in winter but sprout back again in the spring.
Root veg should start to be ready any time now. If you have early potatoes, have a little grope around the roots and see if you can find any potatoes. You can pull out the first ones without disturbing the plant, and leave the rest to grow on. Carrots: we've just pulled the first ones, which are just finger-sized babies, but that helps thin out the crop and we leave the others to ger bigger. Beetroot will probably need more time yet, you'll have to feel around the shoulder of the plant and see whether it feels like a decent size.
Oninons and garlic are ready when the leaves start to die back and collapse (probably August).
Cabbage and lettuce - yes, when they look big enough to be interesting, you can eat them. You can start by eating baby ones and leave some to get bigger.
Mangetout: you have to wait until the flowers die down, and look out for the pods. Ours are only about 2 ft tall, the height varies according to variety. You pick the pods when they're still thin, before the peas swell, and eat the whole pod. We've just eaten the first of ours this evening, also our first potatoes. There is nothing like the thrill of eating your first crops!
Hope this is some help. I am banned from the living room this evening (England are playing Sweden) so might as well come here and chat about gardening. Good luck with your veg!