I want to move my sim to a new homeworld with traveler, but without losing her house in her original home world. I want her to stay there for an extended time, then move back to her original home in her original home world. I want her to age normally while she's away from her home, and I want everyone in her first homeworld to age at the same rate, and if possible, progress in their lives, skills, and careers. Can I accomplish this by setting Treat as Vacation to false, and Ageless Foreign Sims disabled for the original homeworld, or is it not possible?

This sounds like a fine way to achieve that; the only caveat is that I'm not sure if Treat as Vacation is as fully functional as doing a true homeworld switch. I just know that I had some issues with Ambitions careers not working in Barnacle Bay if I only switched the vacation status off- whether that was a fluke, or how many other things don't 'like' being in a vacation world, I couldn't be sure.

What I do know is that it's perfectly safe to test this method and check if everything is functioning as you'd expect, and if not undertake a real homeworld switch (which will still preserve the previous world).
Traveller will do its best to age up the original world when you return, either way. Progression less likely, as I don't think SP arbitrarily assigns these changes but rather they come about from SP pushing those sims to do such things. Which they won't be doing when they aren't being simulated.

When you get used to it, it's really quite flexible!
In my legacy, I sent half of my sims to live in France and used travelling without any sims selected to rotate between them and the other half in Twinbrook. This way, both functioned as a homeworld without much fuss. When I wanted to bring them home, I sent one sim to visit them to add them to the household and return (there are simpler ways, but that way preserved their dreams/wishes).

I assume you know that worlds that aren't played actively don't progress. Passively, you can only accomplish the "aging" part, which will be applied as soon as you go there the next time.
Even if you select the second town as new homeworld, your house in the original world won't be sold automatically. It's treated like an owned vacation home.
What I said is true for moving whole households. With partial households, a lot of different things can go wrong upon return, especially if your sim aged up in the new world..

Yeah, I knew worlds don't progress when my sims are away, but I was hoping there was a way to make it work. I decided to send her on vacation to the other world instead of moving her there.