Ok, so I'm breaking in this engine. The first time I was letting it run at 2500, after about five minutes it sounded like it was starting to stumble or miss. This time it was after about eight minutes. Is it likely that my carburetor just needs tuning? I'll check the plugs once the motor cools down enough for me to get my hands around it to see how they look.

Oil pressure was around 60 psi, and the water temp was a hair below 190, but when I shut the engine off, the water temp went up. That normal?

The water isn't circulating when the engine is off and the surrounding metal heats up the water...New engines can be a little fussy, see if you can adjust out the stumble. What was the cause of the initial problems you were having?

Pretty sure it was just the positioning of the distributor after all.

And then after the first time I ran it, when I went to start it again, it broke the gear in the starter in half.

you said after 8 minutes? is your choke opening up completely? That would definitely cause a stumble. just a thought.
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Red 1967 Ford Fairlane convertible with built 289, 4.11s and T5!

I believe it is. I'll have to check that as well I suppose. At any rate, I'm just trying to narrow this down to the carburetor for sure. I'm sure I'm going to have to get a metering rod/jet kit for this carb to run on this engine, though like I said, it runs really smoothly for the first five or six minutes, then it'll get this little stumble in it.

I guess I need to check the timing after it's warmed up. That could be another cause, right? I only checked it when the engine was still cold. I did have the vacuum line pulled and plugged when I checked it. Also had the port on the carb plugged (or covered, rather).

I would break it in the engine with your distributor unplugged from your vacuum port and both ports on carb plugged. I don't think there is need to run your timing advanced. (if i am off on this please let me know?)

What kind of carb are you using?
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Red 1967 Ford Fairlane convertible with built 289, 4.11s and T5!

750 Edelbrock Performer, electric choke. I'm gonna unplug the vacuum advance, you're right, I shouldn't need it hooked up for this.

edelbrock's are "usually" pretty good right out of the box.

Read the instructions for the carb and make sure the idle adjustment screws are where they need to be. To me it seems more fuel related, and for whatever reason it is getting too much fuel. That is why i think the choke is a a culprit (maybe). Engine warm after 6-8 minutes and that choke needs to be completely straight up.
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Red 1967 Ford Fairlane convertible with built 289, 4.11s and T5!

first of all what size engine are you working on it had better be a built 351w or 351c at least to be running a 750 cfm carb. if you only have a 289 or 302 i would put nothing bigger then a 600 cfm if its a little built or a 650 on a built 302 when its cold it will love all that gas but when it gets warmed up it likes to run a little leaner , try putting a smaller one on for a while an see what it does , the most air you can get thru a built 302 is 590 cfm i dont care how much you gouge out the heads .