I don't like ZoneAlarm due to problem I used to have with it many years ago when I was into techy stuff. No idea what it's like now as I've not used it since and it's dropped from the techy must have lists.
I don't know Avira I use Avast! on most of mine now. Personally I would use Microsoft Security Essential on someone else's computer as it doesn't from memory nag you to register every now and again.
But if they are working, leave them be and see how much they annoy you.
I'd still remove ZoneAlarm... though that may cause more problems than not if history is anything to go by though knowing the problems you had before, the more things the better.
If you don't mind my saying, I would suggest a reinstall of the operating system is sensible on any laptop that's new to you. You can't trust any of the software installed on it.
Agreed - windows rot always sets in eventually, a fresh build, AVG or your favourite AV (I use McAfee VSE but I'm biased) and you should be fine, no need for firewalls or use Windows defender if you must..