I have a work/residence permit in Sweden that is expiring the day before I travel. My new employer is applying for a new permit, but I may not have it (or the card) before I leave for travel.
I will be travelling to the Netherlands, back to Sweden, then Norway and back to Sweden. Should I worry that Sweden will not let me back in? Will they ask to see my card at border control, or is this not done between the Schengen countries?
An expired residence permit certainly does not allow you to stay or travel in the Schengen area by itself.
There aren't any systematic border checks so you would not generally meet any border guard at the border between Norway and Sweden or at the airport when flying back to Sweden. You might therefore easily get away with intra-Schengen travel without proper documentation. But it does not mean it would be legal or that you cannot be asked about your status by the police in one country or the other (even without traveling, incidentally).
Furthermore, some airlines won't let you board without an appropriate visa (Ryanair is one airline that is notorious for that, see previous questions like Travel in the Schengen area after my residence card was lost or stolen) and many will at least check your ID so you would expose yourself to some scrutiny in any case.
EDIT: Your citizenship makes your situation much easier. You can legally travel, leave and enter the Schengen area as a visitor for up to 90 days and this short-stay does not interfere with work permits or long-stay visas. Legally speaking, you could also stay in Sweden on that basis alone (people who cannot stay in the Schengen area without a visa might have other ways to get a temporary permit but as far as I know you don't need one and the advice you received about staying in Sweden would only become relevant if the process took more than three months).
Do take the expired work permit with you to document the fact that you haven't been overstaying (your last entry stamp is presumably much older than 90 days), show your passport if you are asked for ID (I would not volunteer a confusing expired document unless it's needed) and you should be fine. If you left the Schengen area, I guess you could theoretically be refused entry when coming back but travel within the area should not be a problem at all, even if there is in fact a border check somewhere.
I am not sure if the information you've received is completely correct. If you've been inside Schengen longer than 90 days, then technically the 90 days you get for free with your US passport is over and you require either a visa or an unexpired residence permit to cross national borders inside the Schengen zone. It is possible that no one will check your documents or that if they do they will not notice or care, but you should be aware that you may be taking a small risk.
The following info is country-specific and so may not apply to Sweden, but for what it's worth: In Spain, unequivocally a person in your situation requires an "Authorization to Return," which, together with an expired residence permit, is as good as a residence permit for 1 trip outside of Spain. Of 2 times that I have travelled outside Spain with one of these authorizations, I was asked to show it once coming back into the country, crossing from France by train. While flying with an unexpired residence permit, I have also been asked to show my residence permit at least once that I recall, when transiting through Switzerland. I did not volunteer it--the border control official checked my passport, saw that visa printed there was old, and asked how long I'd been in Spain.
