1) No. Is there some reason you'd need the money related to the move? If you are a "nonresident alien"
this article offers some advice.
2) Yes, there's no reason not to have a brokerage account at Schwab, Fidelity, Vanguard, etc. There's no benefit to having a savings account instead.
3) FXAIX is a great fund with a very low 0.02% expense ratio. At your level of savings, you might as well allocate 100%. The only downside is the lack of an options market if you want to get into hedging later. However, you probably don't have enough funds invested to do so economically.
4) Your brokerage will have a research tool for ETFs if you want to concentrate in particular sectors, countries, indices, styles, themes, etc. I like VTI, SPY, QQQ, VO, and VB. These have higher expense ratios than FXAIX, but offer me the ability to hedge with options.