Does a little bit of rain or snow damage a building being constructed?

Like mold.

It can. A lot depends on how it is being constructed and the materials used.

A steel structure probable isn’t going to be affected; potentially a little rust but unlikely if the rain or snow doesn’t stick around for long and if it dries fairly quickly.

Wood is another story; there is depends if the wood is simply cut wood, or compressed, or kiln dried, and so on. The wood can absorb moisture and that can eventually waken it.

Is the drywall/plasterboard is up and gets wet, that is a bad thing, because it will weaken, fold, and potentially fall apart when wet.

Bottom line: it depends on:

1) material affected
2) length of time exposed to rain/snow
3) humidity

The answer is in your question. “...a little bit.”

The answer is no.

My brother was having a house built in another state, so couldn’t keep his eye on it. The builders installed hardwood floors before installing windows. It rained a few times (North Carolina, of course) and buckled and warped his floors. The builder had to eat the cost of installing new flooring.

In Florida we build when there is lots of rain, and usually there is no problem. Other parts of the country build during rain and snow, and don’t have bad problems either, but like anything, once in a while things go wrong.

No, provide it has plenty of time to completely dry out before the walls go on.