Honestly, to me this sounds like normal sibling behavior, extroverted or not. My family was all medium introverts (at different levels), and we did things like this pretty commonly - trying to get into the others' room when the other wanted us out. Some of it was simply a power game I think - being able to force your way into their room literally showed you could - and some of it was a normal lack of empathy at younger ages (frankly, empathy isn't fully developed until well into the 20s for many, and certainly not nearly fully developed at 7).
My parents dealt with it in a relatively reasonable way, I think; within reason they let us deal with it ourselves (we were 3 siblings within 5 years evenly spaced, so there wasn't a massive power imbalance, and it was generally not the oldest vs. youngest, usually some combination involving the middle). When it wasn't something we could deal with on our own, then they reinforced that you shouldn't 'force' anyone to do anything, and to give our siblings space.
In particular, I considered this a normal sibling thing because it wasn't something we ever did to other people - only our siblings. It was something we did in the safety of family (similar to how young toddlers/babies often only act out around the parents, and are perfect angels around anyone else - because they know mommy won't leave them just because of bad behavior).
I think that to the extent you can try and stop this, my parents' approach was generally reasonable (and sounds like it's not that different from yours). 7 is still quite young to empathize with someone with different feelings than yourself, so I would probably not spend too much effort working on that for now; of course talk about it, but don't expect it to work for a while. I remember being in high school and still having a very hard time empathizing with people who had different reactions to me. Instead, hard and fast rules ("If she's screaming, stop doing Whatever you are doing), and trying to solve question #2 is probably best (to take away the motivation).
As far as your 'how do you deal with it': I would suggest as much as possible taking some of this time and making it into daddy or mommy and son time. That's the "easy" fix; I realize you're not an extrovert either, and I know it drives my wife bat-crazy sometimes when the kids are super clingy (she's a true introvert, I'm just in the mild introvert area), but it's something you can do in the short term.
In the longer term, he's going to have to find things he can do himself, or find friends to play with. By 7 he's probably old enough to have friends outside the house and play with them at least somewhat on his own, no? I know you home school and this might limit the automatic friend pool some, but I'm sure you socialize some (vaguely remembering other questions). If this 'introvert recharge time' is somewhat schedulable, schedule 'friend time'/playdates to overlap it. If it's not, then just schedule frequent enough playdates that he gets out often enough for his sister to get her own time often enough.
As far as things he can do himself, he's a bit young for any sort of social online experience, but perhaps he might be able to do some creative writing/play? IE, my son who's probably the same extrovert in an introvert household (but 3) when he's left to his own devices often spends a lot of time making up complicated scenarios where he and his friends/relatives do complicated things. I mean things like, he sets up some buses, each bus contains some of his preschool classmates, and they reenact a recent field trip; or he takes his plane and flies to his grandmother's house, and imagines doing things there.
It's not quite the social fulfillment that playing with others is, but it does seem to fill in that role partially for my three year old when he can't get attention from parents or his younger brother. At seven I could see creative writing take this place better; making up characters and putting them through storylines or similar, and gaining some of the social 'feelings' (endorphins/etc.) that way.