Control4 seems to be a range of systems, one of which is an IR extender. So I am not 00% sure about your particular setup.
If all you want to do is to control the different devices yourself using IR, then this is very achievable with Arduino.
You will need an IR led (TSAL6x00 from Vishay) are really good. Vishay IR receivers are also good. For more than a few feet you will have to build a simple transistor circuit to drive 100->200mA thru the IR LED. (The Arduino can only drive less than 40mA I think which will only get you a very short range without a transistor)
There are 'blaster' cables you can buy to pass the IR thru a cabinet, although just putting the IR LED on the end of a cable is fine. (depends on the finish you want).
Also do a search on other IR extenders, if you can't drill a hole thru the cabinet.
Just get an Arduino, IR LED and IR receiver and play around with your TV devices and remotes first, using the IRremote library. Once you have your first sketch running you will be much better placed to decide on your options.
If you need to control over RF then it may be more difficult to achieve, although it is relatively easy to use Zigbee with Arduino, the interface in your installation may be proprietary. There are many simple Arduino RF projects to search for which are much cheaper than a ZIgbee based solution, although Zigbee quality is very good.
Unfortunately, I don’t know enough to help you with your other questions, but it all sounds very 'do-able' provided you can get access to the signals to clone them onto the Arduino.