In this case you can only use "trust to do". There should be an infinitive, not a gerund after "trust".
"to doing" is only possible if "to" is a preposition, but here it is a particle.
And "trust of doing" is only possible if "trust" is a noun, but here it is a verb.
Edit. Usually, when we have two consecutive non-auxiliary verbs, while the first verb can have any form (past simple, past participle etc), the second verb can only take an infinitive form (decide to do, promise to call) or a gerund form (love listening, hate driving), with a few exception when you have to use a so-called "bare" infinitive (make do, let go).
Sometimes a change of forms leads to a change of meaning (stop smoking and stop to smoke are two completely different things).
Unfortunately, there isn't a simple method to guess which verb is followed by which form of the next verb, although there have been attempts to find a logical explanation - all of which have made the situation even more confusing. For example, there was an explanation in Raymond Murphy's second edition of Grammar in Use, which was omitted by at least the fourth edition (I have never owned the third one).
Similarly, it's often difficult to memorize the prepositions which follow certain verbs. Why do people say in English "listen to music" while in Russian it's "слушать музыку" (/slushat muzyku/, accusative case without a preposition)? Why is it in English "it depends on smth", but "ça dépend de qch" in French (/sa depahn deh/, literally "it depends from/of smth")?
However, it's the only case when "to doing" is possible.
I prefer doing something to doing nothing - there the verb prefer is followed by the preposition "to" and/or the gerund.