I have been reading a lot of/several books since last month

They all sound awkward, but part of that is your use of since, especially since last month in the first two examples. I would replace that with for the last month. It even makes the first sentence sound OK:

I have been reading 10 books since for the last month.

It implies that you have been concurrently reading 10 different books. (Perhaps you're taking 5 classes and you have 2 books assigned for each class)

I have been reading a lot of/several books since for the last month.

Presumably, you finished one and then started the next, but it doesn't have to lớn be that way.

I have been drinking 3 cups of coffee since this morning. (probably wrong)

It means you've been concurrently drinking 3 different cups of coffee this morning. It's not something you would commonly hear anyone say.

I have been drinking a lot of coffee since this morning.

This is good as you've written it. However, it's less awkward if you eliminate the word since:

I have been drinking a lot of coffee this morning.