They often time have problems fitting. You can try to gently peel out the plastic pieces and put them over the square sensors first. The only issue I’ve seen with that is taking those out might cause a scratch on the color paint and it will be visible from the front of the screen. If you try that, use a plastic/nylon spudger to pry those pieces off. Sometimes it just takes some jiggling those square pieces around until you get them in. You may have to push firmly to get them in.
This could be more of an issue with the HDMI IN port. Does it feel loose when you plug it it? I’ve seen a lot of those at our shop recently that are going bad and a new HDMI port fixes it.
Has this phone been repaired before? If it has it could be the proximity sensor; it could be broken or not lined up right. If it hasn’t been repaired before, and you haven’t tried, just turn off the phone and turn it back on. Sounds simple but it might do the trick.
It is possible, but you have to remove the screen, which is pretty fragile. Me and the company I work with usually don’t mess with Surface Pros, they seem to have poor designs for repair. If you want to try it, you can go to eBay probably to get the part. You would need a lot of heat, using a heat plate or heat gun, and tools to remove the screen. Once in I don’t think actually changing the power button is that hard.
I’m wondering if the LCD cable is damaged. I just fixed one of these but had to replace the LCD because I crimped the LCD cable unfortunately. Is it both screens or just the top? It happened to the top screen for me, but sounds exactly like the issue I was having.
Did you notice when it started happening if anything major occurred that could cause this? Let’s try checking the battery cycles by going to the Apple on the top left, Selecting “About This Mac,” click “System Report,” then click “Power.” In the power section near the bottom I believe it gives you a cycle count. If that is high, or if there is a message saying the battery needs replaced, it could be the battery going bad. Also, do you notice if it gets hot at all? It could be an overheating issue, but doesn’t sound as likely if it is shutting off only after 5 minutes.
I have had this problem often, and what I have found is if you connect the port to the board before laying in down and screwing it in, it helps the connection be snug first, before you put it back together. From the pictures it looks like the connector cable is slightly bent, so it’s probably not making a great connection. You may be able to save that piece if you slightly bend it back straight and again plug it in before it’s laid fully down and screwed in. If it still doesn’t work, you may need to start over with a new port and do it this way first. Once the board is screwed in you can’t move the port to where it needs to be to perfectly align with the connection on the board, so connecting it first helps with that.
It sounds to me like the LCD cable might have been ripped or pulled out of it’s connection on the screen. Either that, or the LCD is broken. Before you jump to that conclusion, try just reconnecting the LCD and making sure the connection is totally snug. If the connections are good, unfortunately you might need to replace the LCD.
Has it been repaired or opened before? This also could be the sign of an Audio IC chip issue. There is a chip on the board that controls microphone input and speaker output. This causes issues with phone calls and the voice recording app, but has been known to cause the phone to restart randomly or take a very long time (like 5-10 minutes) to boot up. Does it ever boot up? That is one idea, but it could be the phone. It could be possible to get it in to recovery mode and restore the phone through iTunes, too.
This concerns me because it may be the “No Service” issue. You listed it as the model A1660. This model has a known issue by Apple due to hardware issues with the phone. A chip on the board somewhat breaks off it’s solder points and causes a “No Service” message at the top left. Is that what it is saying? If so, it may be the No Service issue and I would contact Apple. If it says you have service but you aren’t able to make calls, that may be an issue with the SIM from your carrier or an antenna like you suggested. If you can provide us with more info we can try to help you figure out what your next step should be!