The games and achievements I complete on Steam now—do they expire, or do they count toward next year?

Im a little confused by the question, but no games and achievements do not expire. There are instances where something might get pulled from the store for purchasing, but it is not common.

Steam only measures achievements and playing time from January 1 to December 18. But what happens to achievements during this break?

Steam achievements are permanent, they don't work on a time frame. I've got ones from 14 years ago.

Yes, but will the achievments I achieve in the meantime, from December 18 to January 1, be listed in next year's review, or will they be ignored?

I think you're talking about the Steam Replay thing?

It mentions in the FAQ that it includes 'all of your play time between the first second of January 1 and the last second of December 14' for each year so based on that anything you play from the 15th until the new year would not be included in the 2026 one.

That`s what I wanted to know.