Reported Speech "Will" vs "Would"

Solution 1:

NB: I find it nicer to put "to say" before "that", though why I find that nicer is probably a subject for another post. I've done that here.

If the class was in the past then the first example is fine.

Lucy expected you in the class on Thursday. Where were you?
Didn't she get the message? I sent her an email to say that I wouldn't come to the class.

The second can be used when the class is in the past or future. It can also be used to show something that's continuous.

Lucy expected you in the class on Thursday. Where were you?
I sent her an email to say that I wouldn't be coming to the class.

Lucy sent me. Is there anything you need for your presentation on Tuesday?
Don't you know? I've moved to a different town. I sent her an email to say that I wouldn't be coming to the class any more.

If you're using would for events in the future, it indicates that it's likely, unless something changes.

I sent her an email to say that I wouldn't be coming to the class unless I got fired from my new job.

The term will, or its negative, won't, can indicate a stronger, more likely absence, and only works with classes in the future. There's far less possibility of change in this. Here's your fourth example:

Did you hear that Lucy's banned the use of acrylic paints next week?
Yes. I sent her an email to say that I won't come to the class.

Your third example shows an ongoing absence, rather than just one occasion:

Lucy loves the work the students did last week. Did you know she's banned acrylic paints for good?
Yes. I sent her an email to say that I won't be coming to the class.

They're all valid, with slightly different meanings. Generally, if the class is in the past use wouldn't, otherwise use won't, and if the absence is continuous, use be coming, otherwise use come.