short sleeved vests are like underwear (knickers and cami in one really) how many you need depends on how frequent and messy your little one will poo and how sicky they are as sick does get through the top layer. I would have more than 5, at least 7 or 8.
If it is hot a vest alone will do fine, no need for trousers unless you want to.
Leave longsleeved vests for now unless you are expecting baby in october or later. the little buggers grow while you watch
babygros/sleepsuits again depends on how sicky baby is, have at least 7.
They do not have to go in a separate wash load, as long as you use non-bio washing powder/liquid.
If it is new wash before first use.
You will be surprised how quickly that small stuff dries on an airing rack inside or a rotary/line outside. I have no tumbler either (my tiny bit for the environment).
You say no mini-outfits, but after a few weeks you will want to look at shorts and t-shirts or trousers and tops if cooler (at least if like me you have a boy I suppose). Personally I find it much easier to whip of trousers than to undo all those pesky poppers on a sleepsuit when changing nappies.
If you need more vests/sleepsuits get multipacks from tesco, they are cheap but stay in good shape.
Feel free to fire more questions.