I had to enter that mall first and then TAKE (why isn’t it “took”?) an elevator.

The verb that shows time is "to have"

I have to take the elevator every day.
I had to take the elevator yesterday.
I'll have to take the elevator every day in the future.

whWen I first heard her SAY (why isn’t it “said”, or “says”?) that.

When I first heard her say that....

The action "heard" take places in the past.
When I first heard her say that... = the verb "say" is an infinitive
I heard the dog bark
I saw the tree fall down
I watched the man fall down

I had to enter thate mall first and then TAKE (why isn’t it “took”?)take an elevator.

The word "that" calls back to a previously mentioned noun, in this case, a mall. Though, as you are introducing the mall with this sentence, you cannot use "that," and must use "the" which is more general.

Take instead of "took" because, despite the sentence beginning in the past tense, the second half becomes present tense when you're telling a story. To keep this in past tense, you would say "I ENTERED the mall first, then TOOK an elevator. . ."

when I first heard her SAY (why isn’t it “said”, or “says”?) that.

When you use this sentence structure, you are introducing a past action and then transporting the reader into the moment, making it present tense. In general when you begin a sentence with the past tense, you will use the present tense to follow it, though there are exceptions.