There is a sentence like "I met her a second time. or I recorded it a second time." Does "a second time" mean "for a second time"? and if so, is it possible to drop "for" like this? Does this mean,"It was my second time meeting her, or It was my second time recording it."? Can I also say, "I met her the first time."?
Yes - it is possible to drop the "for" in those sentences.
"I met her a second time"
Is the same as saying:
"It was my second time meeting her."
"I met her the first time."
however, does not sound natural.
I would say either,
"I met her for the first time (at a party)..."
"The first time I met her (was at a party)...