Are attack rolls and saving throws basically specialized ability checks? They aren't.

Are saving throws ability checks DND 5e?

So while saving throws and attack rolls are based on your ability scores, they are not ability checks.

Do Ability checks count as actions?

There is no such general rule

Whether something requires an Action to do and whether a successful check is necessary to do it are completely separate questions. There is no rule stating that if you have to roll an ability check, doing so also consumes your Action.