Suggest (like command and require) may take a "that" clause where the verb is in the base form, eg:
I suggest that he do this.
(Some people - more in British than American English, I think - use the present: I suggest that he does this. Historically what I've called the base form is a present subjunctive, but since the present subjunctive is ...
Your advanced student is wrong.
This is an indirect way of suggesting that someone not do that
Above, the subject of the second do in the that clause is someone.
This is an indirect way of suggesting that someone not to do that
Above, what your student said changes do into an infinitive. This means it takes over a noun function, and changes do from ...
To say that something turns out to be the case, you can use 'prove' followed by either: an adjective, or an infinitive verb and an adjective or noun.
The tools proved useful.
The new employee proved to be a fool.
The car proved to have a flat tyre.
Of your sentences, only the first is grammatical. The second uses the wrong verb form after 'proved' (must be ...
"To" is a preposition that can be used to say many things. You can use it to indicate:
A goal [As in your question]
A direction of movement
A place of arrival.
So, your question:
"Can someone help me [with this thing, now]?"
Yes they can...
"Can I get someone to help me [tomorrow]?"
Yes you can... TO is indicating a goal you wish to achieve - ...
No, the sentence is not ambiguous.
To say that something is particularly useful for certain purposes is NOT to restrict it to those purposes.
To say that a hammer is really useful for knocking in nails does not mean that it cannot be used for other purposes. A knife provides an even better example. It makes perfect sense to bear in mind that a knife can be ...
You could replace "to keep in mind" with "to use" and it would be a bit clearer. On the other hand, the way it is written, the speaker is not necessarily advocating that you use it. When arguing with a narcissist, you would remember this previous technique and consider implementing it at an appropriate time.
It does, however, usually imply that you would ...