First off, the first bullet point contains some redundant wording. The following would be more concise:
Answering: Open all gates.
With bare infinitives ("open" and "switch"), your sentences are understandable. However, finite verbs generally provide more information than non-finite verbs, because they can indicate tense, person, etc.
There are multiple ways to make the verbs finite, including:
(A1) Answering: [[This]] opens all gates.
(A2) Ignoring: [[This]] switches lights off after two minutes.
(B1) Answering: [[You]] open all gates.
(B2) Ignoring: [[You]] switch lights off after two minutes.
(By the way, "To switch lights . . ." would not be a complete, well-formed sentence.)
In version A, you can omit the subject with little loss of meaning. In version B, there will be more ambiguity if you omit the subject. I'd therefore recommend version A, preferably with the subject, although you could omit the subject if you really want to reduce each sentence's length slightly.
To answer your last question, dropping the subject is often called "conversational deletion" or "left-edge deletion". You can find more information if you google those terms.