It's very possible to get an STD from a toilet seat or from unclean hands but it's very unlikely unless the disease-causing organisms somehow are able to get into your Vagina.
Which it could if you stick your finger up your Vagina or the orgasm left the toilet seat somehow and got into your Vagina.
It's very unlikely that you would get an STD from your unclean hands or toilet seat though.
Still you should always wash your hands after using the bathroom or touching a toilet seat and use toilet paper on a public toilet seat.