I just recently purchased a new 16" upper, factory built, from a company at the upper end of the "mid tier" of AR rifles.  After the first ~25 rounds (PMC Bronze .223), I opened it up to run a boresnake through and as I separated the lower from the upper, I noticed two small parts skittering across the table (items in the center and right side of the photo).  I wasn't able to readily ID them as parts I recognized, and I wasn't sure if they were from my rifle or just loose junk on the table. An inspection of the rifle by me and one of the range staff didn't reveal anything noticeably amiss, so I shot another 20-30 rounds and everything felt/heard/looked fine.  End of range session.

When I got home, I racked the handle to do a brass check before putting it away in the safe, and a small piece of roller pin (left side of photo) fell out of the ejection part. So loose parts are definitely coming out of this gun. However, I can't tell if something is broken internally, or it's some junk left over from the manufacturing/packing process. Nothing in the gun (bolt, carrier, extractor,  trigger group, mag release, etc) looks broken or appears out of spec to my well-read but admittedly amateur eye.

Lower is a home-assembled receiver with an Ace stock and LWRCi Enhanced LPK which has provided at least 1500-2000 rounds of flawless performance with a variety of Colt and BCM uppers.

Does anyone know what these parts are?

Forward assist pawl.

Ah. Of course.  I typically assemble lowers myself and buy factory built uppers. I haven't assembled a forward assist in years.  

What risks would I be taking to continue shooting this upper without replacing the innards of the FA button? I am competing in a fairly big (for me) match next Saturday, 8 days from now. Not a whole lot of time to get this sorted out and currently living in an apartment with no bench, vise, receiver block, or any upper tools.

There is a spring somewhere also. If you replace it you may want to make sure the front edge of the pawl is not sticking into the upper and catching the notches in the carrier when it should not be. The tip looks like it may have caught on the carrier during the firing cycle.

Replacing a forward assist doesn't require much in advanced tools.  A small brass hammer and a punch will do it.  Start the retaining pin, insert the FA, then tap the pin the rest of the way through.

Or...

Run without an FA.

^This. Easiest solution: remove the FA.

That pawl doesn't look right to me, looks like the end tip has broken off, definitely damaged in the lower corner.  If you replace it prior to your match, check closely that the new one isn't hanging up.  Or you could just run this match with just the fa button in place and don't worry about it, but make sure you remove the spring if its still there. And best check for the broken pawl tip other debris in the FCG pocket.

Contact the company you bought the upper from, this is hot right and needs to be fixed. As for your competition just run it without the FA.

Manufacturer made it right and overnighted me a new full FA assembly. I had the right punch in my toolbox and it took maybe 15 minutes to swap it out.  It took me more time to safely clamp my upper in the vice than do the actual work. I really should go buy a Geissele Reaction Rod.

Thanks everyone for all the help!

Using a Reaction Rod to install a forward assist would be a bad idea.

Installing a FA is the one thing a clamshell is good for

Yep, they work pretty good for that.

Use the dpms panther claw or similar tool that retains the front and rear lugs to Replace/Repair the forward assist. If you're having problems aligning the pin with the forward assist in place, you can try placing a punch on the opposing side whilst tapping in the new roll pin; you'll notice that there's a void for the pin when pushed and orientated correctly. I find it easiest to oil the pin and hole and use a roll punch starter, punch all the way to the last 1mm by a larger punch with one or two blows, and finish with the correctly sized roll pin punch, which has a nipple on top to help center it.

I was gifted a Wheeler AR Armorer's block, darn thing works wonders for many assemble tasks such as holding the FA in place while you tap the roll pin in place.
Wheeler AR Block