find sth to doing?

The word "getting" is being used in this case as a gerund, not a participle and not part of a "to"-infinitive. A gerund is created by using the "-ing" suffix on a verb, which may then be included in a phrase within which it behaves like any verb, but in which the phrase as a whole is then used as a noun. Thus:

Quietly raising lions is difficult.

The word "raising" is qualified by an adverb, as any verb may be, and "lions" is the object of that verb, but the phrase as a whole, "quietly raising lions", is then used as a noun, which is the subject of the sentence—the thing asserted to be difficult.

The phrase "getting off this rock" is similar: "off this rock" is a prepositional phrase acting as an adverb qualifying the verb "get". The whole phrase, "getting off this rock", is then used as a noun, in this case as the object of the proposition "to".

"The key to solving these problems is this: . . . "

That is perhaps a somewhat more typical instance.

The back door to (doing) something is a noun phrase that is

used to refer to an indirect way of achieving something, especially so that people do not know about it and so cannot object to it (Cambridge)

As you can see, it requires the preposition TO just as you say the door to the back garden.

As @MHardy already stated, getting off this rock is a noun phrase functioning as the object of the preposition TO.

Though instances of this phrase being used with a gerund is not particularly common, it is nevertheless encountered. e.g.

My hope and belief is that that was the Government’s intention, because if it is not it is simply a back door to doing away with home education. (Hansard)

Internships: The Back Door to Doing What You Like (military.com)