Given that it was precedent for US presidents to be limited to two terms, though not explicit law yet, there would likely have been some resistance to him seeking a third and fourth term. Aside from just political grandstanding by the opposition, was there any legitimate efforts to disqualify him or impeach him for violating this precedent? Were there questions at the time of whether he could legally run again and did anyone entertain challenging his election in the courts?
Similar questions to this have been asked and answered in this sub before. This is a copy-and-paste of a previous response, with links to some of those previous answers:
The public was largely in favor of FDR's third and fourth term runs. He won both elections in a landslide. The Republican Party tried to make a campaign issue out of it in both elections, with term limits part of their party's platform. It didn't work in preventing FDR's re-elections.
After his death, the Republican Party gained majorities in Congress in the 1946 midterms and made presidential term limits a policy goal. Enough Democrats went along with it to get the Constitutional amendment passed, in part because President Harry Truman wanted a change to the Presidential Succession Act, so it was a bit of give and take in passing the two laws. More information about the history of the 22nd Amendment can be found here - this should offer you details to some of your specific questions.
Further reading:
This post details the circumstances of George Washington's decision not to run for a third term. Summary: it's arguable that he intended to set any kind of precedent. It was Jefferson who was explicit about presidents not serving more than two terms.
This post details how strictly the precedent was actually upheld. Summary: Not very strictly. Several two-term presidents before FDR had explored the possibility of a third term. Few two-term presidents who survived to the end of their second term hadn't explored the possibility.