“This plan covers medical care and lost luggage.”
In insurance, cover means to include a risk or pay eligible costs. This plan covers medical care. means medical expenses are within its coverage. It is a very common verb in insurance discussions.
“date of birth” / “date of assembly”
Date of birth means the day you were born, and assembly means putting something together. Frank, an artificial human made by a doctor, carefully checks whether in his case it should be called a "date of assembly" instead of a birthday. The next phrase, parts of my body, means parts of his body; it does not refer to machine parts.
“I just wanted to check if I could use it here, too.”
Check if ... means to confirm whether something is so. It refers to the date on his ID. Frank explains that he just wanted to confirm he could also use his usual date on this form. I just wanted to check ... is a modest way to give the reason for asking.
“read all eighty pages of the policy”
Here policy means an insurance contract and its terms, rather than a general course of action. Insurance policy is the fuller expression. The representative tells Frank to read all eighty pages before signing.
“I brought two highlighters.”
A highlighter is a pen used to mark important parts of a text in bright color. The punchline is that Frank is not put off by an eighty-page policy; he has even come ready to mark it up as he reads. It matches the thoroughness he showed in A2-23 by reading the forty-page building handbook.