Is there a way in a TemplateForm, to set a button such that hitting Enter while in any textbox of that TemplateForm, will trigger a button's click event, as is available in other environments?
No, there is no such way. However hitting enter will submit the form as long as you have a button whose type is set to submit (which is the default and expected behavior).
In that case, is there a way to set a hotkey to give the button focus?
This article might help you:
Since using the <u> tag doesn't work, is there a way to underline the letter on the button?