single word requests - What do you call a person who refuses to do something/certain things?

It really depends what they are refusing to do and the reasons for their refusal. And the point of view of the person describing them.

For example, if someone refuses to go to war because he has moral objections to war in general, we call him a "conscientious objector". If he refuses to go because he's afraid of getting hurt, we call him a "coward". If he refuses to go because he sympathizes with the enemy, we might even call him a "traitor". Etc.

If a man asks a girl to go on a date and she says no, he might say she is "stuck up". She might call herself "discriminating".

I don't think there's a general, commonly-used word for someone who refuses to do an unspecified thing for an unspecified reason. It depends on the circumstances. A couple of folks here mention "refusenik", I've heard that word but I think it's pretty rarely used, and is most definitely a definition number 2, only marginally related to the primary definition. Definition 1 being Jews who were denied permission to emigrate from the Soviet Union, as @ermanen points out. Note that in the original meaning, a refusenik was not someone who refused to do something, but someone who was denied permission to do something that he wanted to do. Almost the opposite.