I bought a brilliant, highly useful hdmi splitter from kvmstuff.

This will mean that your TV will only accept 1 HDMI signal through that port meaning a splitter won't work unfortunately.
The only other option is to get an adaptor for another port that you do have spare on your TV, like AV or DVI.

There are 3 options for connecting a DVD player to a television. HDMI, Component or VGA. The most common, and best, these days is HDMI which requires 1 cable, called a HDMI cable, between the player and television. These input / output sockets are clearly marked & there is usually about 3 of them on most televisions. HDMI 1, HDMI 2, HDMI 3. The next is component. These consist of 4 separate connections. RED, GREEN, BLUE & YELLOW. Again simply connect like to like. The third (VGA) are RED, WHITE & YELLOW. Same deal, simply connect like to like on player and set. Once one of the connections are made select SOURCE on the television remote and select the connection that you have made. Cheers

Contact any service technician.

Yes. But both TVs will see the same image from the box. i.e. if you're playing a recording on to the lounge TV, then the kitchen TV will show the same thing. If that's what you want, then the answer is yes. You'll probably need to fit a HDMI 2-way splitter and run some extra long cables to the second TV.

It depends what input connection the second TV has available. You cannot directly daisy chain HDMI cable from Apple TV to the two TVs assuming both have HDMI input. But you can purchase an HDMI 1x2 splitter and connect the 3 up provided the 2nd TV also has HDMI. The challenge maybe the distance you can have between the two TVs. Max is 15 meter but you could cascade 2 units to create longer distances.

http://www.ebay.com/itm/HDMI-Splitter-Box-High-Speed-Full-HD-1X4-4-Port-Hub-Audio-Video-1080p-Amplifier-/191120909726#ht_2675wt_1337

This is the best solution for your problem as a 40 year experience with wires etc. Go to the source of your cable signal connector . Open it and clean the terminals with emery paper , sand the end of the cable shiny clean from corrosion , get a new barrel connectors with gold terminals if possible lengthen the ends of the cable , insert and tighten good, use thick rg-16 or rg 40 cable with good insulation ends crimped professionally with a crimpier or buy ready made , not from dollar store. Get a gold plated professional splitter if you need one or two , it will cost about 15 or more each . Make the cable runs as short as possible, using the thick good cables. If needed get a signal magnifier. From radio shack or the source etc. Connect them good and tight . Ground the splitters . Problem solved , you will get crystal clear channel 2 to 7 including 3 and 4 . Isolate the TV cable from other cables and power wires. Vitally

The HDMI splitter is intended to take one source to two displays. You need an HDMI switch to take two sources to one display, a device you can tell which input to select.

no way to upgrade your pre-amp, but you can just use some simple little RCA Y-Splitters on the rear outputs to essentially give you 4 rear outputs to your amp like with 7.1.

however, true 7.1 each signal is unique.

at some point when it comes time to upgrade, check out Outlaw Audio. They make killer stuff at factory direct prices.

http://www.outlawaudio.com/products/990.html

I do not know if a splitter is available yet for HDMI ,but check to see if the Cable box has an HDMI input and if not then there is no way I know of.