The first one and the third one are fine, but the second one isn't grammatically correct.

#1 is good. It makes sense to have problems WITH SOMETHING: "reading the instructions" is a noun clause that acts as object for the preposition "with" #2 is wrong. You are using "reading" as the object (a noun) of the preposition to. That is perfectly fine, but you cannot have a problem TO SOMETHING (although it can BE a problem TO SOMETHING). Instead, say "She had problems TO READ the instructions" and that is perfect, although "a problem" would work better than "problems". The infinitive ("to read") is very flexible. It can be an adjective, noun, or adverb. In this sentence it acts as an adjective to modify "problems" #3 is perfect. The present participle, "reading" can be an adjective, noun, or adverb. It is similar to #2. Present participles and infinitives can often be used interchangeably. Sometimes the meaning changes, but here "reading" and "to read" mean pretty much the same.

1 and 3 are OK. I prefer 1. It is cleaner and more direct.

"To" can be two different parts of speech. It can be a particle of an infinitive verb form "to read" or a preposition of movement (the equivalent of "a" or "para" in Portuguese or Spanish.) It's not serving either of those purposes in #2. It's a nonsensical construction.