Can a cut/bruise at back of tongue cause swollen neck and pain swallowing/yawning? like muscle spasm or affect swallowing glands?

Absolutely: But see your dentist or possibly an otolaryngologist physician (ear, nose and throat (ENT) specialist here in usa). An ENT has equipment to see further back than most dentists, so they may be able to determine if you have an infection or trauma to your throat or surrounding tissues. See them soon.

Tongue Pain: In short, most definitely. All of the muscles in the tongue attach in the back of your mouth unlike most muscles that attach 2 or more bones together. This can happen even after having routine dental work done. It should go away a a few days but if it worsens or does not correct you should make an appointment with either you physician or your dentist since it could be a mild infection.

Swelling: Having a cut or bruise at the back of the tongue can be serious, especially if you are presenting with ongoing symptoms. I suggest that you visit your dentist or an oral maxillofacial surgeon.