Is it safest to double check my new bitcoin address never showed up in the Ledger (the blockchain) before? I just noticed in my Coinbase Pro, I can click deposit, BTC, and get a brand new address every time. It probably is due to &quot;address is very cheap&quot;. It can be just a new pair of private key and public key, and from that, a shorter &quot;address&quot; can be created.
So since an address is a shorter version, the chance of collision is higher than the private key.
Let's say if a person is to deposit US$300,000 or $3,000,000 into that address, would it be safest to double check that this new address never showed up in the blockchain (the ledger) before?
Otherwise, is it true that when there is a collision of address, both that person's private key and mine can move this bitcoin amount to else where?
would it be safest to double check

Yes but the difference in safety, although not exactly zero, is approximately zero.

is it true that when there is a collision of address, both that person's private key and mine can move this bitcoin amount to else where

Yes.
Statistically that will not happen in any timespan you or I can properly comprehend. See related topics in right hand margin of this web-page. Especially Is a bitcoin address collision possible if generating 90 million addresses every 4 hours?