The body of the phone could be bent causing the screen to pop up OR there’s a foreign object that isn’t letting the display sit correctly in the body. I suggest you take it back to the person who replaced the screen and ask them to help figure it out. I’d listen carefully for any additional problems caused by the breaking of the screen.
My grandson has the same model phone and also had a cracked screen. I purchased a new (after-market) one on eBay for $16.71 and replaced it. The phantom touches are likely caused by pressure from the broken glass. You might be able to reduce the pressure by very, very, VERY, carefully picking out some of the pieces with pointed tweezers. I do not suggest this by someone who isn’t prepared. Have a large, flat, clear surface to work on. Something laid out to catch the broken pieces, and maybe gloves to protect your fingers. Also you might try packing tape on the screen then peel off slowly. See if the broken pieces that come off with the tape were part of the problem.
You can buy the OCA glue (sheets) online in many places but if it were me, I’d return the glass and purchase the display already together. Ifixit carries these and used at a very reasonable price. The reason I suggest returning the glass only is that if it is not done absolutely perfect, you wind up with bubbles or worse in your display. iPhone 5s LCD and Digitizer
It sounds like you have a faulty screen. When you touch the opposite area on the screen to make something happen for example, touch top left to make something happen on the bottom right. Does that sound like what’s going on with yours? The only fix is to return it and get a new, new screen.
Randy's answer is probably going to fix it. You might want to go ahead and check all flex cables to make sure they're seated correctly. Also, make sure the power and volume buttons weren't put in a bind when you put the thing together. I have had that happen (I hate being rushed) and had to remove the mid-frame to fix it.
I had the same problem with the fan on my laptop (this same model) but only after it had been disconnected and removed to service another part. If this is a possibility with your machine, try doing as Oldturkey03 suggests. When you get to the connector for the fan, look inside the plug socket and make sure none of the pins haven't been bent. One pin bent slightly prevents the connector from making contact. This keeps the fan from operating correctly and it's a simple step. I always try the no cost simple things first. Hope this helps.