see below under "The mother"
also, it could be that sex or strong emotions in general triggers the thing to appear. From the established canon of the show's universe, Black Lodge spirits are known to feed upon human emotions (although mostly negative ones; i.e. "garmonbozia"). Who knows? Maybe the previous guy that was briefly mentioned was trying to bust a nut on the job, which caused him to see what he saw?

I dunno, he seemed like a one-off character to me, but I could be wrong. I was kinda wondering the same thing about the drugged out mother in ep. 3, personally.

The gold painting would seem to imply that they are to be used for opening of construction projects or something to that effect. Could be related to the Horne brothers' business possibly?

The obvious out-of-universe explanation is a work-around for Michael J. Anderson no longer being involved With Twin Peaks, but otherwise it could be tied to the quote "Next time you see me, it won't be me" from the S2 finale, although it would make more sense for that to refer to Coop seeing the little guy's doppelgänger shortly afterwards in the same episode.
I have to admit though, I kinda chuckled at "I am the arm. This is what I sound like."

Unsure. Could it be a previously unseen part of The Black Lodge? A very different visual motif if so.

I didn't make much of it. It just seemed like he was mimicking the other guy's behavior, like with everything else he said.

Some have theorized that the thing that appeared in the glass box could have been the "mother" that was banging on the door. I don't know though. It didn't seem to have any identifiably female traits from what I could see.
Also, the woman with eyes was played by Ronette Pulaski's actress. Don't know if that's relevant to the plot or just a cameo.

Now that would be something! Although it would kind of ruin the mystery