Behind the curtain behind the curtain
Thanks for the clarifications, Mr. Crawford. They help us not only play the game better, but understand the game better.
Are attack rolls and saving throws basically specialized ability checks?
Almost. Here's why: the distinction between attack rolls, saving throws, and ability/skill checks is very arbitrary. Basically, the designers are saying "we want to draw a line in the sand, to make fighting its own thing, defending its own thing, and then have a separate line for everything else." This isn't a bad thing - it maintains the flavor of the game and supports class definition. A more solid distinction is when you use these rolls. Attack rolls and ability checks are often active. You choose when you use them. Saving throws are reactive, your opponent chooses when you need them. Also, you get unlimited saving throws each round, while you have a finite number of attacks and checks. So it would be more fair to say attack rolls are specialized ability checks, while saving throws are separate from checks.
When you make a Strength (Athletics) check to grapple or shove someone, are you making an attack roll?
Technically, no. Realistically, yes. As DM, I would easily allow a player to use an attack roll to shove someone.
If a character has levels in more than one class, do the character’s cantrips scale with character level or with the level in a spellcasting class?
Fascinating question! It goes to the heart of "what the heck is a cantrip, and why is it in the game?" And the answer is very telling: you don't need experience as a spellcaster to become a better cantrip caster. Which says in turn: cantrips are a character feature designed to make a character feel relevant as long as his level matches the encounter level. Does this fit into an in-game setting - a 15th level barbarian/1st level wizard casting 16th level cantrips? You be the judge.