Short term your credit score will fluctuate a bit (opened a loan and paid it off in 1-2 months, hard inquiry) but after a couple months it will relax after a couple months. Other than that it doesnt really make a difference one way or another

>I guess my first question is, is this a good idea?

Idk if you need "several of them" at different banks, 2 accounts at different banks is typically more than enough.

>If a savings account pays more than a CD, what's the catch?

It depends on why the HYSA pays so much and the CD pays less. In general with HYSA accounts you just want to make sure you know if there is a limit to how much money earns the high interest rate, if you need to qualify for the high interest rate somehow, and if they have any withdrawal limits you need to be aware of.

Bank A might have an HYSA that pays 3.5% APY but only on the first $5000 deposited, and it only pays the 3.5% if you do 10 debit card transactions per month. But bank B might have an HYSA that pays 3.4% without any limits or requirements, you just deposit the money and earn interest

>My second question is, if you open multiple savings accounts at once, do you get on some list

Generally banks care more about checking accounts because they are higher risk. Opening a savings account should be fine in most cases. There will be a record of you owning the account but that is likely it.

The unfortunate truth is if you want to exist in society today then you WILL have your info tracked somewhere. Renting a house? There are companies that track that. Paying a mortgage? They track that too. Opened a bank account? Tracked. But people ignore it 99.9% of the time and everything works out fine.