"Lesson" is usually specific to what is being taught, while "class" indicates the subject of learning. For example, "Yesterday in English class we had a lesson on pronouns ".
Here is another small difference; if you were to say "I'm having an English lesson tomorrow" I would assume you mean one-on-on, like a private lesson, whereas a group studying a subject together at a school is a class.
#1 depends on if you mean small lesson or attending a class
2. First class, then lesson