1. Correct, as long as it's done within 30 days of rolling 'out' of the 401K, and into a rollover-IRA.
2. If you're speaking of investment minimums to purchase a fund (i.e. $10K minimum purchase of VTSAX), then: YES.
However, if you're thinking specifically of VTSAX (Vanguard Total Stock Market Index - Admiral Shares), you can always invest in VTI (Vanguard Total Stock Market ETF) which has no minimum purchase requirements. Continue adding to your rollover-IRA in coming years. Once you invest over $3,000 in VTI, Vanguard will automatically convert VTI shares to VTSMX (Vanguard Total Stock Market Index Fund Investor Shares). Once you hold > $10K in VTSMX, these will be converted automagically to VTSAX.
3. Read thru JHCollin's
Stock Series. He is an unapologetic Vanguard fan, so you'll probably end up at Vanguard afterward. But it's great recommended reading. I have accounts at Vanguard (taxable & Roth), Fidelity (taxable & rollover-IRA), and TD-Ameritrade (HSA).