Were tank destroyer battalions assigned to armored divisions? Would the crew wear the red/yellow/blue triangle armored division patches on their uniforms?

I've been doing some research on this myself for an M-10 build. From what I've found through photos is AFV crews seldom wore anything more than rank on their uniforms. Exceptions to this are out there, I'm sure. But after looking through a couple of hundred photographs of M-10s in action during WW2 I could only find a few Sergeant stripes, and Army group patches.

Thank you, I meant to ask this as well in my original post...seeings how these were open top, did the crew wear the steel helmets or the tanker helmet?

Both. *Mostly* the driver wore the leather football helmet.

BUT--there were patches out there:

But that's Patton, who fairly wallowed in the spit and polish, plus it's him in the turret of a Combat Car pre-war, so not necessarily representative of anything during the war.

Paul

...but...PATTON!

Definitely, TD battalions were attached to armored divisions. In some instances the attachments could last for most of a campaign. For instance the 703rd was very closely associated with 3rd AD. The 704th was (M-18s) was attached to the 4th AD and so on.
My sense is, and think I read somewhere that the armored divisions didn't typically wear there patches in the front lines. Not only were they not exactly "low-vis" in terms of size and color scheme but why help the Germans know which armored div they were encountering when the 3rd was larger than the other four armored divisions in France. If they could tell by the patch then they probably knew the local div TOE but vehicle codes and sloppy radio procedures probably have that away anyhow!
put italicized text here

TD battalions were not usually organic but attached from higher command. So that's why you see army level patches, not the division they're operating with. Though you will find pictures of 703 and 704 with the appropriate AD patch.

An example of the other case is the 644 which was actually split between 1st Army and 8th ID. But the crews appear to have retained 1st Army patches.

If you look real close at crews that look like they're wearing steel helmets you can occasionally see the ear flaps of the tanker helmet underneath. The helmet shell fit over the tanker' s helmet so crew who needed to retain radio communication (tc) would wear one over the other. Loaders and gunners are trained to respond to just being yelled at and drivers respond to being kicked.