New Year is the first day of the year, which for the Gregorian calendar is January 1. In most countries, that is a holiday.
The sentence shown by the OALD ("We are away over the New Year.") implies "We will return after the New Year." Rephrasing this sentence to use be away, it becomes "We are away until after the New Year." This is not different from the following sentences.
They didn't reach the border until after dark.
He refused to comment until after the trial.
You cannot legally take possession of the property until after the contract is signed.
Until means "up to the point in time or the event mentioned"; in "until after the New Year" the point in time is "after the New Year."
The idiom "to check out" has a lot of meanings, but here it means sense 10: to be confirmed. You are performing a check using the side mirror; you are checking the mirror for traffic. The use of the idiom here is fairly informal and imprecise, but not wrong.
The use of "checks" rather than "check" is simply subject-verb agreement with the indicative pronoun "that," which is the subject of the clause.
I check.
You check.
He/she/it checks.
"That" refers to your check in the mirror; i.e., the absence of cars in the driver's blind spot. Basically, it refers to the sentence's entire first independent clause, which is everything before the first "and."
Having answered your specific question:
Wow. The quoted sentence is a doozy. First of all, it is overly complicated and should be broken up into separate thoughts. Secondly, the second independent clause is ungrammatical:
and [...] the next rule of thumb is something that is absolutely mandatory to doing this properly and safely is to turn your head.
After hiding various modifiers, the stripped-down version looks like this:
and the rule is something is to turn.
The author appears to have completely forgotten that they already wrote the first predicate, "is something ...", when they wrote the second predicate, "is to turn...".
Best Answer
The "over that of" is a comparison that could be replaced with "more than" or "better than". Even as native English speaker, I had to read this sentence multiple times to understand it because of how long it is. Stripping out technical details, I've reduced the sentence to get a much easier sentence to parse.