A Few Newbie Questions

[Niwashi]

1. Should I level to max before learning another class?

You should at the very least take a detour at level 30 to pick up the secondary class needed to unlock your job.

Whether you want to do more than that is a personal preference sort of issue. Getting secondary and tertiary classes high enough for all their cross-class skills is useful, (and since you'll definitely want it eventually at endgame, may as well do it early so you have those skills available while leveling as well). Adding additional classes/jobs besides your main gives you more flexibility in what you want to focus on any particular day. Some people like to leave all that for after they get their first job fully leveled. Others like to do several of them side by side. There's no major advantage either way.

2. What is the best way to make moola in FFXIV?

Crafting and gathering. But to do it well, you need to level all eight of the crafting classes and two or three of the gathering classes, so it's a major time commitment.

3. Do I need to do any homework before Satasha?

It's not truly necessary, but you can look up a guide or two if you want an idea of what's coming, like:
eorzeareborn.com/sastasha-dungeon-guide
finalfantasy.wikia.com/wiki/Sastasha
As you get into later dungeons with more complex mechanics, more people will start expecting you to know how to deal with them in advance. In the first few dungeons, people will be happy if you know your class skills and your role.

There's some mechanics even in Sastasha, like clams that can only be attacked when their shell is open, or grates in the floor of the final boss room that bubble when an add is about to spawn, and you can prevent the add spawn by interacting with the bubbling grate. But the mechanics don't have major consequences for failure until later dungeons, so some players just ignore them and treat it as a basic tank and spank.

5. What are linkshells?

private chat channels. If all are highly active, being in several could be rather chaotic, but lots of them sit idle until a conversation gets started. You won't necessarily have simultaneous conversations going on in several linkshells at once. If yours are active enough that that happens regularly, you can seperate them out to different chat tabs to help keep them clear.

6. Does this game have any known issues with 4k display support? Or other resolutions?

I don't know personally, but saw this in a recent Letter from the Producer thread:

4K resolution support
We’ve been receiving comments from players who have high-spec PCs and 4K displays mentioning that the UI is too small. We will be addressing this over multiple patches.