Buying gear online is a pain in the ass not being able to try stuff on, especially gi's and rashguards where fit is important. I use BJJHQ and MMAHQ which sell one item a day, typically higher end items at a decent discount, but terrible return policy. MMAwarehouse and Amazon are also good if you can find what you want.
For the BJJ items:
Gi: if you're training in a gi every other day you should be fine with only 1, that's enough time for most dry. If you're on back to back days unless you get a really light weight one it likely won't be fully dry (you don't want to put them in a dryer). I like Tatami gi's because the fit me well, all I can recommend is find a brand that fits and just use those, different manufacturers cut them differently. I picked up a Tatami gi for $60 on BJJ HQ, if you wait long enough something like that will come up.
Mouthguard: I got spoiled, one of my training partners is a dentist, he made me a custom fit on for free. It's way better than any of the boil and bite versions, there are outfits that will send you a kit to make a mold of your teeth but I have no experience with them, I think they cost a lot too. My moutguard is by far my favorite piece of gear.
Jock strap: Some people wear a cup and some don't for grappling, I do. I use a shock doctor jock w/ cup pocket and then compression shorts over it. There are some compression shorts with a built in cup pocket, but I find that moves around too much, it can lead to bits and pieces sneaking out such that they are in between your thigh and cup, at this point taking an impact on the cup is, well, lets say you want to avoid that at all costs.
Rashguards: You're probably ok with 1 as they dry quickly, unless you want long and short sleeve. Also you don't really need a rashguard under a gi, only if the material irritates your skin.
Headgear: Some people wear wresting headgear for BJJ (e.g. Cliff Keen). I don't and will jokingly make fun of my training partners that do, then again they never have to drain cauliflower ear, so maybe they're the smart ones.