If you have already tested multiple batteries and received the same issue, then yes that can be the problem. Apparently these guys are located in Europe and can replace this IC for you: http://cepmobilservis.com
First, reassemble your phone to make sure you have connected everything properly. It is possible you accidentally unplugged the power/proximity sensor flex cable. If it still doesn't turn on after that, then yes, your next step would be to plug the phone in to your computer and see if it is recognized in iTunes. - If it is recognized, then you have a faulty screen. If you can, try the original screen and see if it turns on. If the old screen doesn't turn on either, then you might have a shorted backlight circuit. - If it isn't recognized, then you might want to take it to a repair shop for diagnosis.
If you can, remove the EMI shields before soaking in alcohol. Then once it is submerged, clean it gently with a toothbrush or something similar. If that doesn't work, I have seen lately that a bad U2 chip can cause this problem. It is difficult to find someone capable of that repair, however.
Well your backlight coil most likely needs replacing. Cyberdoc LLC is very good at this repair. As far as the apple logo goes, a number of things can cause this problem. I encountered recently that a bad power/proximity sensor cable can be the source of this issue. Try disconnecting that cable from the board and start the phone using the charging port to see if that is an issue. I would try this with all the "pluggable" components to see if any of those are the problem. BUT BEFORE DOING ANYMORE TESTING, I HIGHLY RECOMMEND YOU FIRST CLEAN THE BOARD THOROUGHLY WITH ISOPROPYL ALCOHOL TO ENSURE THERE ARE NO SHORT CIRCUITS CAUSED BY CORROSION
Yes, it is integrated with the front camera assembly. Also, you can remove the plastic alignment piece from the original LCD. Pry it off carefully and do it inside of a box or something because it has the tendency to fly off and can be easily lost. Be aware that you won't be able to glue it into the new LCD, you will need to just place it in the correct location and it will be held in place by the camera and speaker bracket.
I had a similar problem with a 4S when I replaced the power/proximity sensor cable. Once I had replaced the cable and reassembled everything, I turned the phone on and it would freeze at the apple logo and sometimes enter a bootloop. I started inspecting the circuit board for damage but couldn't find anything that was apparent, and then I tried unplugging the power cable/proximity sensor. With the cable unplugged, I started the phone by plugging it into a USB cable and it worked perfectly! So I would try starting the phone in that same manner to test the different components. It is possible that a bad component other than the battery or dock connector can be the problem.
If you try and remove the screen with a suction, there is a good chance you can make it worse. I recommend using a razor blade to wedge between the bezel and the bottom edge (by the charging port) to pry it open rather than pull on the screen. Use this kind: http://www.vansonink.com/products/Razor-Blades-%239-single-blade-(100).html Make sure you insert the blade PARALLEL to the edge of the bezel so that the ENTIRE edge of the blade goes into the seam. This way you prevent scratching the frame.