Could sciatica cause testicle pain?

Not common, but yes: Sciatica is a condition where the sciatic nerve is irritated. The nerve is made up of branches from the 4th lumbar nerve (l4) down to the 3rd sacral nerve (s3). Scrotal and testicular sensation are typically provided by the s2 (racepinephrine) nerve. There are other more common causes of testicular pain, but disc herniation in the back can potentially cause this pain.

Yes and no!: This is occasionally the place that sciatica manifests, but it is an uncommon and non-specific manifestation. Most sciatica occurs at the lower nerve roots which shouldn't go to the testes. Testicular pain is often referred from other sources (hip, intestines etc). Technically, if a root is going to the testis, it is not a root of the sciatic nerve and thus not sciatica but this is nomenclature!

Yes it can: Happen. Suggest seeing a pain/spine specialist to evaluate you for options regarding treatments.