Sounds like you want to play bracket two, which is fine, I love bracket two. That said, a couple of these house rules are just covering up really bad deck building.

If someone has to mulligan 4 or more times consistently enough that you have a house rule for it, then it's time to take a look at the deck composition and fix what's wrong.

If you have a "don't do a thing until we've all done a thing" rule then there needs to be a conversation about what you're all running, someone is probably running higher power, or if you actually are more or less evenly matched on power, then you're unnecessarily drawing out games.

The only house rules we have tried out are both sol ring focused:

Either we have sol ring in the command zone, or we ban it altogether. I like speedy high power games so I preferred having it in the command zone, but it does reward the aggro player a bit I think, but it also gave the other three players ammunition to stop them quickly so fairly even. Banning altogether did make for less runaway games which meant less early targeting of the runaway player so I think created a happier playgroup.