I am interested in running a 1 x 9 drivetrain using an 11-speed Shimano 105 R7000 rear derailleur. If I'm friction shifting 9 speeds, will using an 11-speed derailleur cause issues? Or is it as simple as setting the limit screws to the proper positions. Would I need to use an 11-speed chain because of the derailleur?
It won't necessarily just work, and there is a way of predicting whether it will with good accuracy, barring a borderline case.
If it's a Shimano 7, 8, 9, or 10 shifter in friction mode, it probably won't work.
Friction shifters have some variance in their total cable pull between models. It's easy to measure by marking the cable and using a ruler or caliper to note the total movement of the mark.
Take that total cable pull number and multiply it by the actuation ratio of the rear derailleur. For Shimano road 11, it's basically 1.4. That number has to be equal to or greater than the total cog-to-cog distance of the cassette, which for Shimano 9 is about 34.8mm. If it's less, the rear derailleur won't have enough total movement.
The reason there can be borderline cases is because rear derailleur movement for a given input is actually a curve, not a perfect ratio, and the exact start to finish value is going to depend on the high limit setting. But it's pretty close to just being a ratio.
A 7/8/9/10 Shimano shifter in friction mode has a little bit of movement beyond where its last detent position would have been, but not very much. Since these generations are designed to pair with a 1.7 actuation ratio rear derailleur, that small bit of movement is not likely to be enough to compensate for the fairly large difference in actuation ratio.
I don't have any experience putting 9-speed chains in 11-speed derailleurs. Sometimes that direction of mismatch can cause rub between the derailleur inner cage and the chain. 9 speed chains don't tend to experience this in 10 speed derailleurs and 11 speed chains are only something like 0.2mm narrower than 10, so 0.1mm per side. It will probably be fine to run 9 but you could do 10 or 11 to be safe. The shifting may be laggier but it's unlikely to matter with friction.
Nathan is right - the most important thing is that your shifter has enough cable pull to get the derailleur across the cassette. A Shimano 11s derailleur has a 1.4 pull ratio, so to move it 34.8mm you need ~25mm of pull. It should be more pleasant to use with a lower ratio derailleur as the cogs will feel further apart and be easier to trim.
It's worth noting that the sprocket pitches listed are averages - the smaller cogs will feel closer together and the bigger ones further apart as the derailleur doesn't move in a straight line.
Running an 11s chain with a 9s cassette will suck. The outside width of the chain is narrower and it will fall between shifts. Maybe talk to your LBS about running a 9s chain through an 11s derailleur, but I imagine it'll be fine.
FYI my setup is 9s downtube friction shifting with 9s derailleur, but I'm looking at moving to 11s. Trimming is tricky as the cogs are so close together but you get used to it.
I don't thin there will be any problems being able to shift an 11 speed derailleur with a friction shifter.
You need to run a 9 speed chain on a 9 speed cassette. There might be a problem with the wider 9 speed chain fitting in the R7000 derailleur cage.
The only anther problem I see is that you'd be wasting money if you buy an R7000 derailleur specifically for this purpose.