When adding an additional outlet off an existing one with Wiremold, does the new outlet box need to be grounded if the starter box is?

Run a piece of bare conductor from the existing box, through the wiremold, to the new receptacle.

Definitely yes.

Run a ground wire from the existing outlet to the new one. Then the box will be grounded through the chassis of the outlet itself.

If you are using metal wiremold the raceway satisfies the Equipment Ground Conductor (commonly called a ground wire) requirement, but the receptacle needs to be grounded to the box at the next box.

And if you do pull an EGC into the raceway then you need to connect it to the receptacle and the box. You would think the raceway should ground the box, it does seem the gradient or inductive difference in the ground would be negligible, but I don't pretend to know more than the code making panel that governs the requirement, and the NEC 250.148 requires all grounds be connected together and to the metal enclosure at every box that contains a termination.

There are a couple of other scenarios that develop that make little sense, like I once pulled an additional 30A circuit into a 3/4" emt raceway that only had 20A circuits in it. The conduit satisfies the ground requirement, but in my haste I failed to notice that there was an existing (optional) #12 ground wire in the raceway, the electrical inspector wrote a correction notice, I pulled the ground wire out and then it passed.

do NOT trust the metal raceway as a ground. Current electrical code says you should wire in the ground [bare] to the outlet and the box

I'm assuming the wiremold you are referring to is some sort of conduit in this situation. Conduit is no longer considered to the ground alone, hasn't for some time. You are still suppose to run a separate ground, or at least in my state it was required, going on 20 plus years now.

The actual box DOES NOT need to be grounded, but the OUTLET should always be grounded. If this is even REMOTELY news to you, YOU shouldn't be installing outlets.

It depends if the type of raceway is listed for grounding (I believe Wiremold is, but check it; plus the raceway has to be properly connected to the ground). Personally, I'd run a ground wire anyway just to be safe.

Anytime you terminate the Black, White or Red, terminate the ground.