Given that you've got it in a case, it likely will not get worse over time unless you are regularly straining it by removing it and replacing it in the case. I would certainly keep an eye on it anyway. If it does shift to a point where debris can get in under that screen, I'd be concerned, but as of right now, I wouldn't advise a repair, as the risks during repair outweigh the reward of having a flawless frame.
Speaking from a person who works in a repair shop, you should be holding the shop accountable for the quality of the work. If that is not an option, you can use super glue, but it's something that should be regarded as a last resort. There is a risk of brushing glue over your camera aperture, part of the LCD or some other part that is exposed just under the glass. On your S5, I'd probably just be worried about the front camera, ear speaker, or maybe the menu flex at the bottom of the device. Look at the teardown guide and see what is near the area you have to glue, and if you go ahead with this, use as little glue as possible.
You can try leaving it on the charger for a few hours to see if it turns on itself. Sometimes your battery will drain during a repair and it can take awhile to get it back up and running on a normal level. The other possibility is that you got a bad charge port. This happens, especially when ordering parts online or wholesale. I typically expect faults in about 1 out of 20 parts. You may need to return it to the seller for a new one.
I see this happen a lot, and it's usually due to using a poor quality adhesive or from not cleaning the frame of the device before applying new adhesive. There is a bit more pressure in that area just from the bulk of the cable, but make sure you are tucking it snuggly down into the slot next to the LCD and it should rest well. As for the adhesive, some digis come shipped with adhesive on them. I usually find this to be of low quality. I like to use red double-sided electronics tape and a bit of heat under some clamps to get everything to stay down. Once in awhile the digi will still lift right there, at which point some careful application of a tiny amount of super glue (use a little brush) helps keep it snug. The only issue with super glue is that you don't want to get it on any flexes or the LCD when applying it, as it can make removing the digi a nightmare if you have to open it back up for whatever reason.
You likely need a new charge port flex. The charge port, aux port, bottom mic and antenna are all part of the same flex and a repair guide can be found here. The part is pretty inexpensive, just be sure you double check the seller reputation.
Sounds like a software conflict, either with an app or a recent Android update. There are a few solutions here that may work, however factory resets are a last resort. Note that you will lose all data if it comes to this, so back up whatever you need to beforehand.
Try a new battery. When a battery swells, even a little bit, it needs to be removed from the device immediately before it risks damage to other components. Provided this has not already happened in some way with discharging or a short, a new battery should allow you to turn the device on.
Dirt/sand/any small particles that were trapped between the case and the phone. If they were moving enough to leave marks, then they may have also moved enough to have fallen out by now. Could also be a poorly manufactured case where excess flash from the injection mold or a seam in the plastic is rubbing in that area.
Once the glass has cracks, it has weaknesses. There's not really any cure for this. Any sort of adhesive you use is really just going to keep the pieces from moving much or falling out. I have seen people apply clear packing tape with the intention of continuing to use the device since it still worked fine, and if the cracks got worse or further physical damage happened that rendered the screen non-functioning (in this case you might lose touch response), they'd go ahead and replace the glass. Really, any choice outside of full repair of the damaged part is just a band-aid.
You would need to disassemble and reasseble the display and ensure the LCD is seated properly, or that no parts/screws are loose in the area to cause pressure from behind. This may have also been caused from heat applied during the repair procedure, so it may not be pressure.