While we haven’t specifically tried that project with a Raspberry Pi 4, I suspect it should all still work fine.
Other than the additional parts you listed, you will also need six AA batteries; we typically recommend using rechargable NiMH batteries, in which case you will probably want a charger too, if you do not already have one.
The Robot Arm Kit would be a fun addition to that project, and the additional ball caster is a good idea when using the arm to help prevent the robot from tipping forward; you can increase the tension of the front ball caster as needed using some rubber bands. If you don’t have a plan yet for powering the servos, some consideration are mentioned in the “Using the Robot Arm for Romi” section of the Robot Arm Kit’s user’s guide. One option described there is using something like our #2859 D24V22F6 regulator to reduce the voltage to 6V for the servos.
We do not currently carry any Raspberry Pi boards directly. Raspberry Pi stock has recently been hard to come by, although it seems be improving over the last few months; you might check Adafruit or use something like this RPi Locator tool.
We don’t have a specific recommendation for which Raspberry Pi to use (it will generally depend on what you’re doing with the robot), but any flagship Raspberry Pi board starting with the Model B+ or newer (e.g. Raspberry Pi 3 Model B+ and A+ as well as the Raspberry Pi 4) should work with the Romi.
Unfortunately, I do not have any alternative suggestions for incorporating the robot arm.
As far as a battery charger, since the Romi uses standard AA cells, you can use any basic AA NiMH battery charger. I personally like ones like this 8 bay charger that can do AA and AAA independently (some chargers require pairs or certain sets of batteries at a time to work). However, please note that I cannot vouch for that particular brand or seller.
I just ordered the Robot Kit for FIRST and an Expansion Plate and Robot Arm Kit. I forgot to order batteries/battery holder. Does battery holder come with the kit? Which batteries do you recommend, and how many?
I figured out the batteries - just need to know if I need a battery holder.
The AA batteries for the Romi get installed into battery compartment built into the underside of the Romi chassis. For reference, you can see how they are installed in the “Assembling the Romi Chassis” section of the Romi Chassis user’s guide.