When mentioning specific hours, we always use 'at' (eg. at 6pm, at quarter to seven, at 4:30am, etc.) 'On Monday' is correct. You will hear people not use 'on' more in conversation, because it is quicker to say. When we mention 'last, next' we don't use on. In terms of 'giving/teaching/having', it depends on the context (i.e. whether this is every Monday, or specifically next Monday). 'Teach a class' is always correct. ('He teaches a class every hour'). 'Have a class' is also correct, but often refers to both the teacher and student together, or only the student. ('We have a class every Wednesday', 'We will not have class this week'). 'Give a class' sounds unnatural to me. To describe this situation, I would say one of these: She's not available at 6pm next Monday. She's not teaching at 6pm on Monday. There's no class at 6pm next Monday. We're not gonna have class at 6pm on Monday.