Learn English – Allow introducing theself vs. allow me introducing theself

The usual way to convey this meaning is to write "I am writing to ... (inform you/tell you) that...". I've never seen the expression you cite, not in English. I know it is accepted in other languages, maybe non-native English speakers translate it from their mother tongue in English.

Both uses of the phrase are correct.

I would probably use the 'begs the question of whether or not A=B' as it flows better, but the 'or not' is superfluous really. Whether A+B only has two options, true or false, so the 'or not' doesn't really add anything to the phrase.