Can men who had cancer in one testicle have kids?

Yes. The remaining testicle can still be functioning. A sperm study can be done to see if there are any issues. The other treatments like radiation and chemotherapy can impact this but may also leave sperm function unaffected.
No reason why not. Except that those with teticular cancer tend to have low sperm counts, even before diagnosis, but certainly after anesthesia. Checking shortly after may show low numbers and weak swimmers. If going to chemotherapy, makes sperm banking an issue. Counts and fertility can change with time.
Yes and no. If you had radiation and or chemo, it may have done permanent damage to your ability to make viable sperm! if no significant damage was done and you had a simple orchyectomy then you have good chances of being virile.