Television and Home Theater

When TV is plugged in there is no picture or sound. Just a loud buzzing sound.

Recently my tv has been not been showing a complete picture. It has lost about a good inch on all sides of the screen. For example when playing a game the word "WIN" will pop up on the screen but now it displays "IN". I took it in to Sears for repair as it is under warranty until Aug 26 2006. They claimed to have resoldiered the V prom and it was ok for a day and it is now doing the same thing again. In fact, after they "fixed" it, it takes longer for the colors to fully display compared to before being taken in.

We have a Sharp model 19L-M1DDBS and are not able to get some cable channels. For channels above 47 we get a solid blue screen but if we moves the cable connection in the back of the TV the channel comes in. We tried a new cable, still have the problem. Seems the connection inside the TV is loose ?? I'm wondering if this is worth fixing or just buy a new TV ?
Also, the remote goes from 2 to INPUT to 2 when pressing the channel button, instead of going through all the channels. Can you tell me how to resolve the remote problem ?