Try pulling the RAM out and plugging it back in to see if it gives you the beeps that it recognizes that the RAM isn’t there. Do you get any kind of POST chime? If not and you have reset the SMC which it seems you have, disconnect the HDD and Optical drive from the logic board and see if it keeps doing it, my hunch is that it is a good chance that it is the logic board. If there is a local Apple shop around I would take it there because an Apple Store won’t touch it because they consider it obsolete.
Did you replace just the LCD or did you replace the entire front panel (Digitizer and LCD)? You might need to reseat the cables on the logic board or if you bought the digitizer and LCD combo, you might have gotten a bad part.
The RAM that is compatible with you laptop is PC-10600 (1333 MHz) DDR3 SO-DIMM, I recommend Crucial or Corsair brand RAM. Amazon has 8GB Apple OEM kits for $51- https://www.amazon.com/SAMSUNG-10600-SODIMM-LAPTOP-MEMORY/dp/B0067GGPXC/ref=sr_1_10?ie=UTF8...-10&keywords=PC-10600+%281333+MHz%29+DDR3+SO-DIMM, or you can max it out at 16GB for right at $100. Here is a link for a 16GB kit - https://www.amazon.com/Corsair-Certified-Laptop-Memory-CMSA16GX3M2A1333C9/dp/B006ON5KZC/ref=sr_...-8&keywords=PC-10600+%281333+MHz%29+DDR3+SO-DIMM
Actually there is an adjustment screw that you could try before replacing the trackpad, if you have the laptop open, it is at the bottom center, right below the trackpad, you can see it in the video thumbnail. It is a T5 screw. Try that first before replacing the trackpad.
I would suggest never going back to the repair shop that did not even open the iMac and told you it was the LCD or the MLB. I am an Apple Consultant and also repair only Apple computers, so I have been in quite a few iMacs. Usually ends up being the backlight board but always check the cables because I have had 2 that I can remember where another repair shop and even the Apple Store had done a repair not related to video issues but messed up the v-sync cable. The v-sync cable is very delicate and you can easily mess the pins up or crimp the cable causing intermittent or total loss of backlight. I know you have solved your issues but thought I would add this for anyone else searching on how to fix this problem.