Vostro 3668, GTX 1070, compatible?

I believe your Vostro 3668 has a 240W PSU. The 1070 requires a 500W PSU so it would require a power supply upgrade as well. You could go to a GTX 1050 though as users have installed that card with success. Just make sure of the card dimensions. The mini version of the 1050 card should be fine.

It wont ever work and it wont ever physically fit inside the case.

Power supply is proprietary and not standard physical size.

I would Recommend a GT1030

PNY  VCGGT10302PB

https://www.pny.com/geforce-gt-1030

Dell A9763687

https://www.dell.com/en-us/work/shop/accessories/apd/a9763687

https://topics-cdn.dell.com/pdf/vostro-3667-desktop_owners-manual_en-us.pdf

DELL 3470

OPTIONAL CARD