Heathrow to Gatwick

Yes. Both immigration (for you) and customs (for things).

Yes, these airports are rather far apart. There's a direct bus you can take between them.

And you will need the UK ETA.

You will be responsible to move your bags between airports.

How much time have you got between flights? If your inbound flight is on time I would want at least 5 hours between flights. If you are on unlinked flights (probably) and your inbound is late or cancelled the second airline will have you as a no show and cancel your return too and you will have to buy all new tickets at walkup price.

Welcome to the Forums.

By the way, you posted in the Beyond Europe Forum. After Brexit the UK is no longer in the EU but as always was, is in Europe. There is an England Forum where folks from England hang out and answer questions.

What Nigel said.
Please be sure you have a good few hours to transfer between airports.
The distance doesn’t look far, but it is.
Pre book the coach/bus transfer if you can.
Remember that roads in the UK are busy and you may experience delays due to traffic.
How much time do you have between flights?
Is there a reason you have booked flights in this way?
Can you change them so that you are staying in just one airport?

We had this kind of ticketing with BA but it was the other way around....from Athens to LHR with a flight to the US from Gatwick. I did not realize the 2 airports were that far until I was looking into how to transit. It was not too much of an issue for us because we stayed overnight in Gatwick. am not sure if that was just that particular day but the lines to check in in Gatwick were out to the door and real chaotic! It was a total nightmare! If that were still the case, you definitely would need time to make your connection. That was in March of last year.

It's now much easier to get between the two airports by train than it used to be - Elizabeth Line from Heathrow to Farringdon station, then Thameslink to Gatwick Airport.