But there's a feeling every classes for 25 and 50 mins

'every classes' => 'every class is' ?
'there's a feeling' - I don't know what he means by this.
'Im' => 'I'm'
'And' => 'and'

That's not important. He is disrespectful I can say for sure.
It is like teaching etiquette while picking the nose.

8/10?! seriously?
i'd say my English is 8/10 -- and theirs would be somewhere on the wrong end of the chart

** с энда начинать - это дурной тон по всем стайл-гайдам **
That was true a few decades ago. At grammar school in the 1970s, for example, we were told not to start sentences with conjunctions ('And', 'Or', 'But' etc). However, it's now increasingly common in a lot of standard written English. Even BBC newsreaders now start sentences with conjunctions, and have done for 30+ years – but that's because, in the 1980s, they began to adopt a more relaxed style of reading. I also begin sentences with conjunctions from time to time. It's definitely still not acceptable in technical or legal documents, though. If you're in any doubt, it's best not to use conjunctions at the beginnings of sentences.

Наша задача ____(не сократить, а вообще убрать) разницу в зарплате между мужчинами и женщинами.

It depends on the style of language used in the original. If it's a transcript of a conversation or is a text written in an informal/relaxed/colloquial style – like a lot of the postings on this forum – it's not a crime to begin the occasional sentence with a conjunction in English. People often do this deliberately to create an effect.

However, with more serious styles it should normally be avoided – and you should definitely avoid starting sentences in official/technical/legal/'serious' texts with conjunctions, because it will make the text look too 'chatty', which will make it lose some of its authority.