I think it depends on the circumstances: A lecture sounds like talking in front of a congress or like a professor "teaching" the students. In cases of one student/pupil talking to (a limited number of) other students/pupils I learned to use the word "talk" instead of "lecture". But generally I think there are nicer ways to start a talk of whatever kind, like just starting to tell an anecdote or some recent events suiting your topic, if your time frame permits you to. (Still I only learned English and someone who speaks it natively might have a better idea about it!)

It depends on the situation (doesn't it always)....Since you used the word "today" I'm assuming that this is for a class that meets regularly as opposed to a one-time lecture. Your introduction is correct and suitable for a university professor's introduction to her lecture. This is the kind of thing people say when they're talking to a large group. There are some other ways to introduce a lecture that might be more inclusive... "Today we're going to look at...." "Today I'd like us to explore / examine / etc...." or you could jump right into the matter... "You've probably heard the term "dark matter" before and thought that it had to do with that leftover food that's been sitting in the back of the refrigerator for the last month. But dark matter is actually much more fascinating and much less dangerous than anything you've got in your refrigerator..."

Today's lecture is on.....

Your sentence is correct, there is nothing wrong with it. You could say Good Morning/ Good afternoon or Good evening first.

good sentence