- I can't see anything on that board that would be worth the time or effort desoldering...
- Any chance of getting that LCD to work with the Arduino is likely low, unless you can obtain a datasheet for it
- Do you have desoldering braid, or a solder sucker (plus picks and such)? Without those, desoldering parts will be at best an exercise in frustration...
Regarding the LCD - any chance of using it likely went out the window once you started removing wires. I be very curious as to what is on the other side of that PCB; I'm also curious as to where the controller for the LCD is located at (if it is a part of the LCD PCB - that's better, but you're gonna want that connector for the ribbon cable). You need to identify what the LCD part numbers and such are (you might also try doing some googling on the LCD TV model number itself to see what comes up); if that fails to turn anything up, then the part numbers and such for the controller ICs are the next step. Another problem you're going to run into is not knowing what the voltages and such the LCD and backlight expect (this is why, before you start tearing thing apart, you need to do research and maybe throw a scope or meter on parts if they work at all, while taking a ton of notes, and pictures, and labeling things). Even with all of that, it may just be a bunch of dead ends; the parts may be OEM/house marked and made or marked exclusively for Thompson (Panasonic?) and impossible to decipher (and forget about getting help from any company today).
I'm curious what you expect to be able to do with any of the coils on the board (that you couldn't do with other means)? I mean, what do you expect to -do- with one of those inductors (I guarantee you it won't work well as an electromagnet)?