I would like to start working on a project which involves controlling the temperature and flow of water in a pipe, by regulating the flow of warm/cold water into it.
In order to do so, I need to be able to measure temperature and water flow in both cold and warm water pipes, and to regulate the flow in both.
So what is the real objective? It sounds like you are just trying to regulate temperature by mixture. If that is the case I don't think you need to measure the water flow. Arduino just measures the temp and adds hot water as required.
I've seen solenoid valves, flow meters and thermometers, but do you know of any product which combines the three? (I'm talking about household-type piping, so that's more or less the scale)
The individual components are very cheap. If a combined device is available, it will probably be quite expensive.
If there is no such product, is there a relatively cheap thermometer designed for measuring water in a pipe? (I'm trying to avoid buying a normal thermometer and drilling it into a pipe, if that's possible)
You can hardly go wrong using a DS18B20. It is only a few dollars, accurate, has no need for calibration, is well-supported on this forum, and very easy to use.
There is no need to drill the pipe. You can use a thermowell, which is normal practice for pressure systems, or simply lay it on the pipe under an insulating sleeve in some thermal paste or even silicon. If your pipework is Pex, it would be a good idea to let in a short section of copper and fix the sensor to that. This would make life easier for the Arduino.