Are Buddhist monks allowed to hug other people?

First for context, Buddhist monks generally aren't big on the whole punishment thing; sure some things like killing, stealing or sex, fundamental Buddhist precepts, are essential to be upheld for the integrity of the Sangha, but an innocent hug is not going to get you disrobed, as many particularly in our culture see it as an act of compassion or loving kindness.

Sure it is against the rules and against tradition to touch monks in certain more conservative parts of the world like Thailand, particularly with the opposite gender, a good monk and lay student will be mindful of this, but I am sure that many do not mind very much, I think the typical "punishment" for petty things such as this is to explain to the head monk or other monks, and they will advise or warn as necessary.

His Holyness, the Dalai Lama sets the example for millions of Tibetan Buddhists:

I would sure hope so. Been hugged by a great many monks. Some of the most friendly people I have ever met.

It depends on context. Generally monks are not to touch women and Buddhist nuns aren't supposed to touch men. However, when I was in Thailand I saw a nun who was having a hard time walking on the sidewalk because it wasn't level. I offered my arm for her to hold onto and asked to carry her bags. She grabbed my arm, for safety, and I walked her to her destination.

not in Theravada traditions - as much as I know. shaking hand is also not appropriate. it can be hard in western world. ask the knowledgeable monks for certainty.

read here http://www.buddhanet.net/pdf_file/bhkkrule.pdf

Generally no.