Is there a way to automatically add the current date to the lower right side of my placed signature?
Which kind of signatures are you talking about?
If you have the cryptographic digital signatures (Certificates -> Digitally Sign) on your mind, you can create custom appearances for them including the signing time..
Digital signatures automatically show the date (and time) they were applied, and if they don't, that information is saved within them and can be accessed via their Properties. Any other kind of signature will not have such a feature built-in. One could create a stamp with a script that does it, though.