Brad

Do the usual and start from the beginning. Reset, hold both Home and Power buttons until the iPhone begins to turn on, usually less than 20 seconds time. If nothing do the same while on Wall Charger, again Reset. If nothing, then with iTunes open, hold Home button and connect iPhone to computer, keep holding home button until iPhone is recognized in Recovery Mode, usually less than 20 seconds time. Then Restore iPhone. If nothing, perform DFU Mode Restore.

See step 31: https://d3nevzfk7ii3be.cloudfront.net/ig...

Those look like pressure points, most likely being applied to the back of the screen. If this is a replacement screen, it may have previous damage that is just now showing up. The fix is a good front screen assembly, and just make sure everything behind the screen area is in order, so this does not recur.

Connect the iPhone to the wall charger for about 10 minutes. It may turn ON itself. If not, keep on the charger and Reset, hold both the home and power buttons together for about 20 seconds until the iPhone begins to respond.

The home button just sits loosely in the gasket. The sticky side of the gasket is on the same side as the touch surface of the home button. Just line everything up perfectly, (the wings of the home button go to the sides, and the square on the front of the home button, is perfectly square like a square, not a misaligned diamond) with the touch surface of the home button facing out towards the touch surface of the front screen, and press it into place from the back of the screen.

In step 25 photo, this piece appears at top center of iPhone 4S frame. https://d3nevzfk7ii3be.cloudfront.net/ig...

Along the lines of Jake's thinking, it could be you left the power flex under the logic board. Open up the iPhone and check your work. The power button flex cable connector is possibly under the logic board. If so, carefully take the iPhone apart and lift the logic board so it can be routed on top of the logic board and connected properly.

That is the back plate and bracket of the front screen assembly. The next step is from the outside of the iPhone you need to unscrew the two pentalobe screws next to the lightning port. Then you can carefully open the iPhone, you must be very careful not to tear the home button flex cable as you lift the front screen bracket.

Just find any cracked but working iPhone 4 or 4S screen to use. Like you say, you don't have to install the cracked screen just connect it.

Both of those APN are correct for iPhone 4S. The connector on the new battery should be exactly like the connector on the old one. If the connector is different you got a fake or mislabeled battery.