湿る

You can use 湿る (shimeru) like this: 雨で洗濯物が湿ってしまった。 (The laundry got damp from the rain.) Or, 梅雨の時期は部屋が湿りやすい。 (During the rainy season, rooms tend to get damp.)

湿る (shimeru) means to become damp or moist, usually not completely soaked. Think 'humid' or 'a little wet'. 濡れる (nureru) means to become wet or soaked, like being drenched in water. If your clothes get a little damp from humidity, that's 湿る. If you fall in a pool, you're 濡れる.

Yes, 湿る can be used metaphorically. For example, 空気が湿る (The air gets humid). You might also hear it in expressions like 目が湿る (My eyes are moist/teary), though that's less common for 'damp' and more for 'wet with tears'.

Yes, the kanji is 湿. So the word is written as 湿る (shimeru).

You can say 空気が湿ってきた。 (Kūki ga shimetekita.) Or simply, in a weather context, 湿気が多い。 (Shikke ga ōi - There's a lot of humidity) is more common.

The polite form is 湿ります (shimerimasu). For example, 洗濯物が湿ります。 (The laundry gets damp.)

Yes, you can. For example, 湿気でクッキーが湿ってしまった。 (The cookies got damp/soggy from the humidity.)

湿る (shimeru) is a godan verb. You can tell because the dictionary form ends in 'ru' preceded by an 'e' or 'a' sound (in this case 'me'), and it conjugates like a godan verb.

The past tense of 湿る is 湿った (shimatta). For example, タオルが湿った。 (The towel got damp.)

While not a formal idiom, you'll often hear phrases like 空気が湿る (the air gets humid) or 心が湿る (your heart feels damp/gloomy, meaning you feel down). The latter is a bit more poetic.