Is small nodules that look like skin tags on the genitals always hpv?

No. Very commonly these are genital warts caused by hpv, but they could also be molluscum, skin tags, seborrheic keratoses, or another lesion. Most of these are benign, but when in doubt, always have these examined, especially if they are changing or growing.
No. Small raised skin changes aka papillomas have different origins. Skin tags. Hpv and molluscum bodies are common findings.
Genital lesions. They can be hpv, they can be tags, they can be other viral lesions as molluskum.