Gmail Search Operator for general “today”

I don't know of a direct way to do this, but you can create a filter that marks any mail containing "Re:" with a specific label (lets say RE), and the search for -{label:RE}

alternatively, skip the labeling part and search for -{"re:"}

Kinda, there is a work around to do what you want, while doing the search, you can specify that results should not have some text, by using the field Doesn't have, and include in this field some text from the signatures.

or if you can also specify to search for whole sentences only, by using double quotes like this:

"my one long sentence"

so it will match the whole sentence only, and not single words.