Learn Dutch: Zal ik je oppikken voordat we bij het diner aanschuiven? - Shall I pick you up before we join the dinner?

Zullen acts as an auxiliary verb here. Whenever you make an offer with Zal ik... or Zullen we..., the main verb (here, oppikken) is pushed to the very end of that clause in its full infinitive form.

Could I use ophalen instead of oppikken here?

Yes, absolutely. Ophalen is very common and slightly more neutral. Oppikken is a bit more casual, similar to saying 'grab you' or 'scoop you up' in English, but both work perfectly in this sentence.

What does aanschuiven literally mean?

It literally means 'to slide up' or 'to push up' (aan plus schuiven). In Dutch, it is a very common way to describe pulling up a chair to join a table for a meal, or joining a meeting.

Why isn't the separable verb aanschuiven split into schuiven we... aan?

Because it is in a subordinate clause. The conjunction voordat (before) forces all verbs to the end of the clause. When a separable verb goes to the end of a subordinate clause, its two parts stick together as one word.