It might be better to just say, "Thank you." and omit "in advance." I think this implies that you are grateful that they took their time to consider your request. It would probably be a good idea to thank them again afterward, this time for whatever work they did to help you.
Unfortunately, some people (many of whom are very outspoken) will be offended by almost anything.
Usually, the best practice is to use the conventions that are generally accepted among whichever group of people you are communicating with.
Emails are not like phone calls; they are more like letters.
With email, the recipient can see your email address or, in some cases, your name before reading the email. (For a letter, you would add a return address on the envelope.)
Just like with letter writing, it is typical to sign your name at the very bottom of an email. If you enable the option to automatically add a signature (which most email apps have), it will be added at the bottom.
The exact way you sign will depend on context.
If you're writing an informal email, you might write only your first name, first initial, or not include a signature at all. Chances are, your friends recognize your email. With informal email, you have a lot of freedom all around, but most people will expect a signature to be at the end (if you include one).
I also don't bother writing my name (or anything much) if I'm sending emails during a conversation with the person. Usually this comes up when working on separate computers, and a file needs to be transferred.
There are two different levels of formal email, at least in my mind.
If it's more casual, such as an email from me to my professor, I will sign it with just my name at the bottom (and I also exclude the "Dear" at the beginning). I may or may not sign my last name, depending on how well they know me (since my name is unique).
In a really formal email, such as one to a company with a job application, I sign with some variant of:
Sincerely,
Name LastName
You can read more about formal email writing here.
Best Answer
You'd not typically use such a sentence (with either this or that) as part of a reply to the mail in question. There would have to be something special about the timing or content of the mail itself.
An appropriate usage would be if someone sent a message out to a number of people, including you. Then if you replied personally to them, you could quote the message and say something like:
With quoting in context of a current conversation, you can use either "this email" or "that email" (all mails can be thought of as being in the past, unless the mail is describing itself). If you aren't quoting it or are discussing something from the past, you'd prefer "that email".
In either case, the wording will stress something unusually eloquent or timely about the email and the choice of when and how it was sent. So it's not something you would use casually to mean "Thank you for sending an email."
Here are some more common ways of just saying "Thanks for writing":