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.
In English reported speech the present tense in the actual words spoken becomes the past tense in the words when they’re reported. The reported forms of those two examples are ‘He asked me what kind of computers I had’ and ‘She asked me what music I liked.’ The situation has changed. The speaker reporting the speech is referring to something that was the case in the past, if only in the very recent past.
As others have said, this is not invariably so. As the authors of the
‘Longman Grammar of Spoken and Written English’ write:
The original speech or thoughts may have been in present tense, but
past tense is usually used for the report . . . Although this use of
past tense in reported speech is common, reported speech also occurs
with other tenses. [The present tense emphasizes] that the circumstances
expressed . . are still continuing.
But if in doubt, change present tense in direct speech to past tense in indirect speech.
Best Answer
The expression does convey a sense of want, but it can take on several nuances, depending on the context. For example:
sounds like a conversation between two friends, trying to schedule the day's activities. However:
conveys a longing for an ability to swim competitively, not necessarily a desire to go swimming that day.
As Barrie said in his answer, sometimes I would like to sounds more polite and less demanding than I want to. I also think it's more likely to be used in the realm of the hypothetical, like in the second conversation above. I think if the speaker was talking about concrete plans, instead of hopes and dreams, the expression I want to might be more likely to be used:
In that case, using want to instead of would like to shows more resolve and determination.
Although these expressions are almost interchangeable in what they mean fundamentally, the subtle nuances can be complex. Even which word that gets stressed can cause a slight shift in meaning:
For example:
seems to convey a different kind of hope than:
Getting back to your question, though, there's nothing grammatically wrong with the sentence you asked about.