Learn English – “relax” or “relaxing”? — I hope you are having some rest and relaxing at the end of the year

Either will be understood, and can be correct, but in my opinion the inclusion of the word "with" has a connotation of a more general problem.

There is a problem sending this form.

This probably implies that there is a specific problem with the form, that prevents it from being sent. Perhaps you forgot to include your last name, and the web site is rejecting it as invalid.

There is a problem with sending this form.

This could suggest a more general problem, which is preventing sending the form, or which may result from sending the form. It may not be a problem with the form directly, but perhaps your Internet connection is not working. Or perhaps your computer has caught fire. Or it may be that if you submit the form, a problem will arise--perhaps your employer will discover you are looking for work elsewhere.

with could also suggest a personal or moral objection, rather than a practical, problem:

I have a problem with going to strip clubs.
She had a problem with the way he sang.

Both of your original conversations look fine. And yes, you can combine your responses into one. Here are some possible candidates.

For example #1:

• "Yes, I have already sent that email (out) today."

• "Yes, I had sent that email (out) earlier today."

For example #2:

• "Yes, I have already sent that email (out) yesterday."

• "Yes, I had sent that email (out) yesterday."

• "Yes, I had sent that email (out) last week."

• "Yes, I had sent that email (out) when I was in Japan."

NOTE: In your excerpt, there is "You CANNOT use the Present Perfect with specific time expressions such as: yesterday, . . ." <== But that is not really true. Your excerpt is an overgeneralized "rule" which is not always true. There are some conditions when past-time adjuncts (such as "yesterday") can be used in a present-perfect construction. For instance, there is a counter-example in CGEL, page 144, [13.ii.a] :

• "We've already discussed it yesterday."

ASIDE: Here is a post with some info on the present perfect construction (though, unfortunately, it probably isn't directly related to your question) : What's the meaning of this present perfect sentence?

NOTE: CGEL is the 2002 reference grammar by Huddleston and Pullum et al., The Cambridge Grammar of the English Language.