Learn English – Don’t go close to him vs Don’t get close to him. when talking about keeping a physical distance from someone

Go not is archaic. It's ungrammatical in Modern English, but people are familiar with this construction as an archaism and will understand anyway. To put it another way, it's grammatical, but the grammar it's part of is that of an older form of English.

Here's a simplified timeline:

• In Middle English (1150-1500 AD), the post-verbal negator not developed, at first appearing with and then replacing the older pre-verbal negator ne. (This is an example of the Jespersen Cycle.)
• In Early Modern English (1500-1700 AD), the modern version of do-support gradually developed, appearing more and more frequently with negative declaratives, with not following do rather than the main verb.
• In Modern English (1700 AD to present), do-support became obligatory in negative declaratives with no other auxiliary verbs, probably roughly sometime in the 1800s, effectively restricting verbal negation to auxiliary verbs (Do not walk rather than Walk not).

Afterwards, the narrow distribution of the reduced form n't allowed it to be reanalyzed as an inflectional affix, which is particularly noticeable in inverted sentences. The affix -n't inverts along with the auxiliary, but the independent word not does not, so Doesn't he walk corresponds to the uncontracted form Does he not walk, not to *Does not he walk—although to this day Heavy NP Shifts result in a similar structure from time to time.

*Mention not is ungrammatical for another reason as well. Mention is transitive, so in the older grammar it should instead be Mention it not, corresponding to the modern Don't mention it.

Of course, if Mention not is really that common in Indian English, then there's nothing wrong with using it. But when I searched, I was unable to find any examples in Corpus of Global Web-Based English (GloWbE); I found only one result listed under Indian English, and that example was a false positive.

The phrase "get to do something" implies to have the opportunity to do something. So the meaning of the first sentence is "I love to have the opportunity to see her first thing in the morning".

The phrases "get to do something and get to doing something" aren't interchangeable to convey the same meaning. The latter phrase means "to start doing something such as I got to talking to her. I got to thinking that it was his fault.

Both the sentences are grammatically correct, but they convey different senses.