I’m trying to activate my free 1-year offer for Apple TV . I purchased and activated my IPad in July 2020

You only have 3 months from the date of purchase / activation of the iPad to redeem the offer. You are no longer eligible for it.

https://offers.appletvapp.apple/

If someone doesn't redeem the offer, will they receive a new offer with a new Apple purchase of some sort? I had no idea I was eligible when I bought my Apple TV last year and the offer expired. Now I'm going to buy a new MacBook - will I be able to get a free year this time?

Yes. If you never redeemed the offer, you can redeem the one from the new purchase.