I have not dealt with Savage service but the following is what I expect from a mfg warranty service center:

As to the 3 week wait, I would guess that either 1) there are very few people working on warranty issues or 2) they have a ton of warranty work ahead of yours.

And, no I would not expect any communications from them if it is a warranty issue. They will fix it and ship it. If you aren't home when the first delivery attempt is made you will have an opportunity to be present the next day, call and arrange a delivery day or go and pick it up from the local distribution center (or post office).

If they determine there is nothing wrong with it then they will probably just ship it back as well.

If they determine there is something wrong, not covered by the warranty, then I would expect them to call or email and ask you if you want to pay them to fix it or for them to return it as is.