About 15 or 16 years ago my teammates and I in school had to design a digital PBX and voicemail system. We used a number of Mitel and also Motorola chips for various telephony interface/features. These days Motorola doesn't seem to make chips any more. They sure make a whole lot of cell phones today... but not the chips or they don't sell it. Then the Mitel semiconductor division eventually became Zarlink of today.
Maybe go to Zarlink.com -> Products -> Telephony -> Caller ID. They may have a chip that you may be able to use here. Then wire the serial or equivalent data to a PIC microcontroller.
Try the PIC 18F family for the microcontroller. You guys can program it in Assembly or C with a free compiler. The ICSP device programmer is affordable and is easy enough to use.
Use the PIC microcontroller to send the name and phone number display data to a 16x2 or whatever equivalent text/LCD display.
The same PIC chip can hook to an infrared encoder chip to mute the TV. For this infrared encoder chip, try Holtek Semiconductor or other companies. Go to holtek.com -> Products -> Remote Controller section. Email them to ask which chip can be use to send infrared data to mute various brands of TVs.
Then look elsewhere for the sequence/codes to send a mute command. To make it "universal", send multiple codes (sequentially; one after another) for the top 5 or 10 brands of TVs and these codes probably should mute half or more of the various brands of TVs out there.
Good luck and have fun.