Half of what I am going to say is speculation on my part, and the other half is knowledge I got here and there, mostly from the excellent Tofugu article, and some from when we were reading Zenitendou with the bookclub and someone gave us an interesting intro to onomatopoeias (I’ll try to find that and edit this post tomorrow with a link to it, when I have more time to look for it).
So, regarding the origin of さっさ. I believe this comes from the sound of feet running, usually represented by さっさっ. The fact that this sound represents running feet eventually evolved into the idea of something happening promptly or quickly.
I kind of picture this as an emperor saying “do this!” and a servant would go running “さっさっ” immediately to get it done.
Now as to what the と represents. When an onomatopoeia can be applied to a verb, it’s pretty common to be connected to such verb by the と particle.
This is different to onomatopoeias that affect other types of words, like onomatopoeia that work like adjectives (example, ほかほかの 肉まん, where the ほかほか is the onomatopoeia for “steamy” and is modifying a noun, like an adjective would).
Now, as to why と is used. My understanding is that this と that can get attached to many onomatopoeia is probably a derivate of the quotation particle と. It’s like how in English you could say something like 「With a “whoosh!” the wind suddenly opened the door」. You can see how English can quote onomatopoeia sounds too.
So さっさと部屋をきれいにした would be something like 「In a “sassa!” I (quickly) cleaned the room」. Obviously “quickly” is a way more abstract concept than the sound of the wind, so the translation looks weird at best, but the idea is the same. It’s just that the Japanese have managed to imagine sounds for things way better than other cultures I guess.
As to whether it’s possible to see さっさ by itself. I have never seen it used by itself ever, but my personal experience is hardly proof of anything. However, I can understand why the dictionary would separate さっさ from と since と is actually a particle, even if using them separately is extremely rare.
EDIT: Here’s the post from the Zenitendou bookclub I mentioned before where a bit of how onomatopoeia can be broke down is explained: