Make sure that you are buying the home button for the correct model. Different iPads use different parts, and if you accidentally ordered the wrong part, it is not likely that it will work. There is not a one size fits all replacement for every model iPad, and each model of iPad is quite similar. The easiest way to identify your iPad is by the model number, there is a tool on the apple website to do this, and once you do that, match the models on the iFixit website and order from there.
The DC in board could Part of the problem, but really the most important thing is to look all around the logic board for any corrosion. If you find any, look to what it does, does the trace connect to a battery monitoring chip? Does the pin on the chip connect to the DC in board? Look all over, even on the other side of the board, for any corrosion at all. My 2009 MacBook Pro was completely wiped out by a little bit of corrosion smaller than a crumb, so it is definitely possible that the power delivery circuit, which is on the edge of the board, was damaged.
Let me get this straight, you are trying to buy an iCloud locked phone and put in the parts from the other phone to make it one phone in good condition? If that’s the case you should be able to just swap the logic board, but apple likes to link hardware to boards. Usually the hardware is still usable though. You can try, but it might be cheaper to just get the screen and housing rather then buying a whole phone.
Make sure the battery is plugged in secure. Its not the charger if wireless charging doesn’t work either, but if the battery charges wirelessly and not wired then the charging port is also not working, try the same thing. It could also be that something else is bad in your phone, since it seems like this was not just a battery replacement for preformance and/or battery life, because you also put in a charging port, what prompted the repair?
I am not 100% sure about this but the phones display cable might have gone loose when the phone got cracked (assuming it was an impact or drop), if you’re ok with a cracked screen maybe take it apart with one of the guides on iFixit and just plug it back in or get a new display assembly and do the same.