Learn Hardcore Norwegian: Jeg leser avisen mens jeg venter på bussen. - I read the newspaper while I wait for the bus.

In English you distinguish read (simple) from am reading (continuous). Norwegian has only one present-tense form, so leser covers both meanings.

Why does the verb venter follow jeg rather than appearing at the end in the clause introduced by mens? I thought subordinate clauses put verbs last.

Norwegian subordinate clauses still use subject–verb–object order. The verb-second rule applies only in main clauses. After mens, you simply write jeg (subject) then venter (verb).

When should I use mens instead of når? They both can translate to while or when, right?

Use mens for two simultaneous ongoing actions (while). Use når to ask or state when something happens, often for single points in time or events.
Example: Når kommer du? means When are you coming?

Why do we form avisen and bussen by adding -en instead of using a separate word for the?

Norwegian marks the definite article as a suffix. Common-gender nouns take -en in the singular.
So avis (a newspaper) → avisen (the newspaper), buss → bussen.

Why are both avisen and bussen in the definite form here?

They refer to specific things: the newspaper you’re reading and the bus you’re waiting for. In Norwegian you indicate that specificity by using the definite form (suffix -en).

What is the function of på in venter på bussen? Do I need it?

vente by itself just means to wait in a general sense. When you specify what you’re waiting for you use vente på + object. It literally means wait on/for.

Can I drop the second jeg in mens jeg venter på bussen, like in informal English?

No. Norwegian requires an explicit subject in each clause (except in imperatives). You must keep jeg in the subordinate clause.

How would I say the sentence in the past tense?

Change both verbs to past:
Jeg leste avisen mens jeg ventet på bussen.
Norwegian has no separate past-continuous, so simple past covers both simultaneous actions.