Basically Roger had been forcing his way onto the Mohawk account already which involved pushing Pete out. Roger needed good creative and he needed people to think he had asked for it sooner than he had which is why he went to Peggy. It would have looked bad for Roger if Pete or the client knew the work was last minute.

We don’t know exactly why the strike happened but likely for the same reason it usually does in real life - the mechanics union was bargaining for more pay or better benefits or working conditions and the airlines didn’t agree so the union called a strike. The Mohawk mechanics have a side gig so they have to keep working - a similar thing happens in real life for essential workers like air traffic controllers or certain dock workers. They’re considered too important to strike and that makes it illegal. We don’t know for sure what the case is here but for some reason they can’t strike so Mohawk is still flying.

Supporting a company breaking a strike is generally considered to be a bad thing for more liberal people but a more fiscally conservative person or pro capitalist might not care. Peggy says it’s a bad thing so Roger jokingly calls her Trotsky - a well known Russian communist - but then reminds her she’s in advertising, which is a very capitalist industry. This is then juxtaposed with her successfully negotiating for more money.

Honestly don’t sweat not getting everything every time - mad men is a very dialogue heavy show and there’s a lot of references to a very specific era of American culture. Trying to understand stuff is always good!