Trolleybuses, as well as so called "shared taxis", which are indeed small buses going on pre-defined routs, are not common to many countries. This fact creates misunderstanding and confusion among the DL community. It would be very helpful if the creators of this unit used pictures to introduce the words for those means of public transportation, which are unique to Ukraine.
More correct is "Ми їдемо у(в) тролейбусі" (if you talk about where are you now) or "Ми їдемо тролейбусом до... вокзалу/лікарні/etc" (if you want to tell where you are going and how). "Ми їдемо на тролейбусі" literary means "on the top of trolleybus" - this phrase in Russianism and unfortunately is widely used now in Ukraine.
Yes. But according to https://rus.stackexchange.com/questions/21188/%D0%9F%D0%BE%D1%87%D0%B5%D0%BC%D1%83-%D0%B5%D0%B7%D0%B4%D1%8F%D1%82-%D0%9D%D0%90-%D0%B0%D0%B2%D1%82%D0%BE%D0%B1%D1%83%D1%81%D0%B5 "на автобусе" became even more common in Russian. Possibly it is not a Russianism but the same common mistake in both languages.
There is a good information regarding the correct usage in Ukrainian here (with detailed table for means of transportation): https://www.kyivdictionary.com/uk/grammar/uk/consulenza-linguistica/vypusk5/na-avtobusi-avtobusom/ As you can see usage of "на ..." should be avoided in most cases. The most correct usage is "автобусом", "потягом", "літаком" etc.