If one is arrested and booked but not convicted will it show on their background?

First its not a good idea to piggyback your question on someone else's. This can lead to confusion. You should start a new thread. So I've moved your question to its own thread.

It depends on the check and how thorough. Usually only convictions will show. But some places may do a more thorough check that can show arrests.

In NY state, a pending arrest will show up on a BR check

In NY, an arrest that led to an acquittal, dismissal will not

However, dismissals and acquittals will show up on a LE check

in NY state, a pending arrest will show up on a BR check

in NY, an arrest that led to an acquittal, dismissal will not

however, dismissals and acquittals will show up on a LE check

It matters who is providing the check, not what necessarily what state you are in.