LCD is a fake or not?

OK, that is normal. I've changed many iPhone 5 and 5s screens before. This usually happens depending on if the screen comes with the front camera holder plastic cover preglued to the LCD to make the camera sit flush with the front display assembly glass camera hole.

Another way is that the plastic front camera holder is taken from the previous one and aligned so that it looks like it's centered as close as possible. This isn't something too much to worry about as long as it isn't making the front camera photos bad.

Not all new iPhone 5 / 5s screens come with a preglued front camera plastic holder for the display assembly.

You can open the phone and use cotton bud to clean the front camera glass and front camera lens if you want by following this guide:

iPhone 5s の正面カメラとセンサーケーブルの交換

Since the screen has been fixed before, it should be easy to get the front camera flex out as it won't be glued to the display assembly, but still you should follow the guide if doing so. The step where you need to heat it up is not required.

Does the phone screen go dark when taking calls and putting the phone next to your ear? If not then the proximity sensor has gone faulty.

For that blue proximity sensor, I don't think I've seen that before and is probably due to the quality of the front display assembly glass. But then again nothing to worry about if the phone is functioning perfectly fine. You are probably right, the rubber gasket for the ambient light sensor is probably missing.

Here are some examples of what the front camera flex contains:

iPhone 5s:

iPhone 6: