oomis:
As far as I am aware, R’ Moshe determined that a tricycle is a toy and permitted while a bicycle is not and is prohibited.
A vehicle, though technically muktzeh, may be leaned upon as it does not sufficiently move (touching is never assur), though jumping on the back of a pickup may be a problem.
I don’t know of any heter to push a baby carriage lichatchila without an eiruv.
Uvdin dechol is usually determined by an action or use of an object which is normally done or operated in an way that would be assur on shabbos is forbidden on shabbos even if the issur is not done. This can be avoided by a significant heker. Example: even if a toothbrush can be used in a muttar way (no water, brush gently), one should set aside a special shabbos toothbrush to avoid uvdin dechol.