Yes, it's called SIPC and is (kind of) equivalent to the FDIC for banks. However it only covers up to $500k, only covers cash, stocks, bonds, CDs, and (I think) mutual funds. It does not cover options or future contracts.

Also it will only attempt to return the securities you 'lost' not their cash value. If you lose a lot of money due to not being able to sell your stocks for several days, that is not covered. So if you held 2000 stocks of GME, the SIPC will (eventually) give you back 2000 stocks of GME, not the cash value of those stocks when Robinhood went bankrupt.

It covers less than $500K if SIPC itself fails (after $5 billion in payouts it is insolvent).

True, it is still an open question if the government would step in in that case or not. As it stands they have no obligation to do so.

You can transfer your stocks to another brokerage without having to sell them.

You can. However, it tends to be a rather manual and long (days to weeks) process.

Your holdings in cash or your holdings in shares?

Both. I suspect there is some kind of FDIC-like insurance but not sure of the specifics.

Yes, it's called SIPC, you're fine unless you have over $500,000 in robinhood.

With the (potentially huge) caveat that any losses you suffer due to not being able to sell your stocks in the period where you SIPC claim is being processed is not covered.