Yes STDs can live on surfaces but only for a short amount of time.
STDs have the ability to live on the outside of the body for around 1 hour or less but any longer than 1 hour the STDs would have a hard time surviving.
That's why it's basically impossible to get STDs from toilet seats, towels etc.
But it's still good to disinfect surfaces such as toilet seats and also use a toilet seat cover on public toilet seats because you could get other infections that may live on the surfaces.
I always bring disinfecting wipes with me to public restrooms and wipe the toilet seat down before sitting on it and I also add some toilet paper to the toilet seat.
For an STD to live though it needs to be in the membrane to survive longer.