It's a little bit stilted. Perhaps most natural and polite would be: "Are you available for an English lesson today?" or "Would you be free to train me on English today?" I think "invite" is usually used if you are the one hosting. Since he is hosting the lesson, it makes more sense to politely ask about his schedule. You are making a "request" instead. But something like "I request you to train me on English today" still sounds very direct, so it is typically better to phrase your request in the form of a question.

It sounds like "it all" encompasses the whole movie-going experience. "The inconvenience of it" - Here it sounds like going to the movies is one simple task. "The inconvenience of it all" - Here it sounds like going to the movies is an experience encompassing several hours. It sounds more like an an ordeal that leaves the person tired. Either usage would be fine. I don't like going through airport security because of the inconvenience of it all. Here you wait in a line and perform several tasks in quick succession (show ID, take off your belt, remove shoes and coats, empty your pockets, remove your toiletries, place laptops on the tray, walk through the scanner, get patted down). Any one of these tasks is not extremely burdensome on its own, so I don't think of going through security as a series of individual complaints, but the overall experience is very unpleasant.

I think the difference is subtle. Native English speakers like me don't really think about this much, so grammar specialists will be a better reference than native speakers. Here's an actual grammar reference that might help. In this case the difference is between a Type 1 (real) and a Type 2 (hypothetical) conditional. http://www.ef.edu/english-resources/english-grammar/conditional/ In your example case using "will" indicates a higher level of commitment. Because you are using a future tense (will), you are indicating that this is something that probably will happen. "If I go to the dance with you it will be fun." "If I went to the dance with you it would be fun." If you use the first option, it sounds like you are actively considering going to the dance. Like you really want to go to the dance. In the second option, it sounds like a purely theoretical thing that would be fun, but may or may not ever happen. "If I try the new recipe it will be healthier than frozen food." Because you use the future tense here it sounds like you may have already made up your mind up and are ready to go. The key idea is that you have transitioned from considering something as merely an abstract concept to building a potential plan for future action. You haven't 100% committed to going, but it's now a "real" situation instead of purely "hypothetical". If I tried the new recipe it would be healthier than frozen food. Here you are not giving any clues as to whether you plan to do this. But as I said native speakers do not think about this much, so the difference is more of a subtlety and they may occasionally use the two interchangably. Additionally, since you use words like "I suppose I could", and "probably" it adds a lot of uncertainty to the statement, so the level of commitment to actually try the new recipes is not very high in either case.

These sentences are fine, but a couple of changes could make them better. To sound completely natural my personal preference would be for: I had a happy childhood. I've told you many times not to park your car there. "Many a time" sounds a bit uncommon or old-fashioned to my American ears, although it should be perfectly grammatically correct. "I spent a happy childhood" would be more common when combined with a location or activity, like "I spent a happy childhood on my father's farm".