There's plenty of ways to do it. The simplest would be to quintuple the size of the house and elect five winners to every district. Literally nothing else would have to change. Five member districts are considered the smallest that are functionally immune to gerrymandering efforts.

A more reasonable suggestion is to start implementing these reforms on the state and local level, where referendums are possible and you have an easier time building a big enough organization to actually get shit done.

As for Approval vs RCV, the simplest answer is that they usually agree on the results [all the way down the line]https://electionscience.org/commentary-analysis/super-tuesday-deep-voting-methods-dive/), but approval is simpler and easier in every respect. Both systems tend to produce a candidate support graph that looks like exponential decay in real life. The complicated answer gets into voting theory/math and all sorts of technical criterion. While I think those arguments are valid, most poling and real world data seem to show that basically anything other than "choose one" is good enough, so I prefer the method that is easiest to explain to voters and hardest for candidates to claim shenanigans.