Train Travel from Frankfurt To Wurzburg and Wurzburt to Berlin

If you have a car reserved, shouldn't it be held even if you are an hour late? Europecar at Wuerzburg Hbf closes at 19:00 and the Sixt at 20:00. Call ahead with your ETA when you get your train sorted out.

ICE tickets do not have to be bought in advance, only if you want to save money with train specific Sparpreis tickets.

Why not pick up the car the next day instead? What will you do with a car that evening?

Short answer - No, it is not reasonable. You would need to get an S-Bahn train from the airport to Frankfurt HB, 14:17 arrive 14:28
Land nominally 13:35 + plane could be late landing (0-60 mins) + taxiing and disembark (15 mins+) + immigration (15-30 mins) + walk to station = no way.
The earliest train you could hope to get (no guarantees) is ~15:00 from the airport:
Frankfurt(M) Flughafen Fernbf 15:36, ICE 721 (no changes), Würzburg Hbf 17:01

Where are you staying that night? Wouldn't it be easier just to get the first train there when you get out of the airport, and pick up the car the following morning. No stress.
Or pick up the car direct at the airport.

"Wurzburg to Berlin. We take the ICE correct? Prepurchasing tickets is needed?" ICE, yes, but you don't need to know that up front. Just go to the DB website ( https://www.bahn.com/en/view/index.shtml ), plug in Würzburg to Berlin, date and time, and it will tell you the quickest train. You don't need to buy your tickets in advance, but it will be a lot cheaper if you do, on the DB website. You will get print-at-home tickets.

If you have a car reserved, shouldn't it be held even if you are an hour late?

No guarantee for that - some car rental companies are rather strict about the pickup time. I second the suggestion to move your car rental reservation to the next day to account for possible flight delays.

Concerning the train connections: When looking up a connection on bahn.de / bahn.com, always enter the places from/to which you actually want to go. If you want to start your journey at Frankfurt Airport, enter "Frankfurt Airport" (or "FRA", which is also accepted by the system). I think there is a chance to make the 14:38 connection if you are lucky and fast - but would not count on it. For that trip, the flexible fare for the ICE is 42 Euros - not much more expensive than the usual recommendation (Rail&Fly ticket, which you should still have a look at if your airline offers it).
I would choose the ICE for comfort and speed - it is not that much more expensive. Consider reserving seats - 4.50 per person, independent from the ticket. If you install the DB Navigator app on your phone, you can even reserve seats while waiting for your luggage, if there are any unreserved seats left by that time. The ticket is indepentend of the seat reservation.

Thank you all for your advice. I’ll see if we can push the rental pick up back. We need to pick up our car rental that day because we are not staying in Würzburg but traveling a short way southeast to Frickenhausen Am Main. We didn’t want to drive from Frankfurt right after getting off the plane.

Ah, that makes sense. Maybe you can just ask the rental agency for how long they are going to honor your reservation in that case.