How do I remove unwanted space over my signature?

You can't the way you showed. The only option I found is to create your own template and to use it for new and replies etc. I found this annoying and reverted back to signatures but it may be OK for you.

Even better is probably Quick Parts:

1. Create a new email
2. Add the signature exactly as you want
3. Select all the text in the editor
4. Click on Insert and under Quick Parts, save

Now when you create an email, select Insert > Quick Parts and choose the signature.

A very hacky way (for new emails) would be to use Auto hot key, and essentially send

1. ctrl n
2. tab
3. tab
4. tab
5. tab
6. delete
7. delete

The above assumes you have To, Cc and Bcc showing (as well as subject field).

You can only remove the spaces that come after your signature, not the spaces before. Your signature files are located at: C:\Users\Username\AppData\Roaming\Microsoft\Signatures. There will be 3 files:

YourSignature.html
YourSignature.rtf
YourSignature.txt

They can be modified and the spaces that comes after the signature removed as already stated above. However even if they don't have spaces before the signature, when you write a new email, the spaces before the signature are added automatically.

The only way I can think of is to try to modify the template Mail.oft. But I am not sure If you will be able to remove the spaces from being added before the signature.