Will the USA be able to tighten up on who is able, to run for president in future elections? Surely they can’t let Trump run in 2020
Nope, the Constitution says that you must be age 35, a natural born citizen, and live in the United States. It doesn't even specify that the candidate not have a criminal record, so even if Scrotus were to be convicted of criminal conspiracy against the US tomorrow (blissful thought), he could still run in 2020.
we've just reached the point at which Pence can serve out the rest of this term & still be eligible to run for two more full terms.
The Constitution limits Presidents to two elected terms (8 years). But if a person becomes president other than by election (POTUS resigns, removed, or dies) and serves less than two years of that term, they can be elected two times and serve an additional 8 years.
It's never happened, but it could. Johnson served out Kennedy's term then was elected on his own in '64. He could have run in '68 but chose not to. The only POTUS to serve more than 8 years was FDR, and he was the reason they enacted term limits for POTUS.