There are two sides to this, one is the taharah/najasah side, and the other is the cleanliness hygienic side. It's always best to avoid public toilets, maybe for both reasons, but if you must, then you have to evaluate if you must stay in state of taharah for salat. If you do not need to stay in state of taharah for salat, then you can use the facility without concern about the first side, becase you can get home, wash, change yourself, do wudhu' and pray.
If you need to stay in the state of taharah, you will need to have a bottle of water, and once you relieve yourself, you can wash.
As for the toilet seat, if you must sit down on it, you can cover it with toilet paper, and if it was wet, try to dry it first, and then cover with toilet paper.
As to whether the toilet seat is facing the qiblah, or its back is to the qiblah, if you do not know, then you do not know, and relieving yourself will be permitted. If you do know, sit on an angle.
It does seem to be a big task, but it isnt. As long as you are prepared, and kind of need to.
And Allah knows best.