Can a green card holder open a Schwab investment account as a normal US citizen would, or most they open an International account? Funds to use would all be coming from a US bank and not overseas.
Your bank doesn't care about your immigration status, it cares about your tax status. You're a US tax resident and will open a US-resident account, not an international account (regardless of where the money comes from).
As a relatively recent nonimmigrant visa holder (O1), I was able to open an ETrade brokerage account without problems. I have full tax residence in the USA so have an SSN, and a credit history so it was no problem.
Later, as a greencard holder, I opened IRA accounts with them, too. Again, there were no issues as I had all the information that the IRS paperwork required at hand.
As a permanent resident in the U.S. but not a citizen, I was told by a representative at Scottrade that I am not allowed to open a brokerage account.