When there is a herniated disc compressing a nerve, do the muscles innervated by the nerve become tight and weak?

YES. The main problem here involves progressive muscular atrophy and weakness secondary to the nerve compression, unless this can be reversed. This kind of issue often results in a decision to do operation to relieve the problems. The tightness, or spasms, may involve paraspinal muscles which contributes to an additional component of pain.
Spasm & Fatigue. Nerve root compression caused by a disc herniation will cause the surrounding muscles to tighten, trying to protect the nerve. The tightening will ultimately "fatigue" those muscles so they become less able to protect the nerve from further injury. This cycle of injury, spasm and tension as the back becomes less able to protect itself from further injury, can cause feelings of weakness.
Yes. Numbness tingling are sensory symptoms weakness is the motor symptom pain usually precedes this weakness is somewhat of a late finding decompressing the nerve root should be considered to facilitate motor recovery.