[SERIOUS] Questions about Ledgers and eventual selling and cost basis calculations for taxes

I have a small amount of coins on Binance and Coinbase (<$2k), I am wanting to move it to a ledger rather than leave it on the exchanges. I have been reading around this subject and I came across a post somewhere that pointed out that if you move your crypto to cold storage when you eventually come to sell you won't be able to find out your cost basis come tax time and (in the US) you have to pay 100% of the value and lose everything. So obviously I'm wanting to avoid this.

My questions are:

1. Does the Ledger live software know your cost basis from details on the block chain?
2. When I eventually transfer my coins back onto an exchange to off ramp them to fiat will the exchange know the original cost basis so my statements will be accurate come tax time?
3. If I put my coins onto a Ledger, make a permanent record of my private key to pass down to my family to rebuild after my death, how will they avoid having to pay 100% back because they won't know the original cost?
4. For such a small amount, should I leave it where it is and not play with it unnecessarily?

Thank you for reading this, I appreciate any help I can get, I'm feeling a little over my head with this but just want to make sure I don't lose what I have.