“Happy Holidays” is typically used when you aren't really sure what holiday someone celebrates. In that instance you are replacing Christmas or Hanakkuh with the word Holiday making it a proper noun, which means it should be capitalized.
I don’t think it’s rude to do the deletion, but it’s pretty rude to not give some sort of Plan B besides “Wait until I decide I am ready to deal with you.” I’m sure it feels lovely to set up if you’ve usually got a lot of annoying people clamoring for your time on matters that aren’t nearly as urgent as they think, but to not even offer a “in case this is urgent, contact X” fig leaf just shows you don’t care.

Picking the right words for you holiday out-of-office emails can seem like a proper “task”, particularly if you want to be both clear and polite with your message, while sprinkling it with usual phrases about the spirit of holidays. Yet, leaving your customers without this type of message is a big no-no, as it presents you as an unprofessional & sloppy equivalent of the notorious Ebenezer Scrooge. Thus, you need to fire up your email client and announce your absence with style by sticking to the following templates and tips.
As for this one I think it’s fine for internal particularly if someone can “read it in her voice” and knows she’s quirky but I’d probably just do a short one for external (or none? because I’ve heard there is some kind of security risk with them?)