Audio, headphone jack, and home button not working

Wow, seems like you've got a lot going on there. I'm frankly puzzled regarding these seemingly unrelated failures all occurring together. The thing is, the home button could be explainable by having stressed the flex cable when you were opening it up; to be honest that connector is a crappy design that just begs to be damaged when you're down there trying to fiddle with that stupid clip holding the connection. If that's what happened, it's easy enough to fix by replacing the home button, but unfortunately that's at the expense of your Touch ID function since only Apple can restore that.

With regard to the audio and headphone functions, that one's a mystery. Assuming you followed the iFixit Battery Replacement guide, there's no need to fuss with any of the parts down there at the bottom of the phone. The two only have a couple of things in common; one, they're both audio signals generated on the logic board, and two, they're both connected to the motherboard through the lighting cable assembly. I'm assuming when you say that audio doesn't work that you're referring to the lower speaker; is that right?

Did you have trouble getting the battery out, by any chance? As in, the stretchy adhesive tabs broke and you had to work the batteries away from the adhesive to remove it? The only thing I can think of if that's the case is that maybe you dislodged the lighting cable connector since it's right next to the battery. Is that one of the things you checked when you went back in and looked it over? If not, it would be worth opening it up and unplugging and resecuring that flex cable on the motherboard. Of course, the taptic engine and the lightning port are also on that cable, so you might think those would be affected as well, but it just depends how the connector is sitting in the socket as to what might work and what doesn't.

But yeah, motherboards generally don't just lose functions at random as a result of changing the battery, so that would really be last on my list of suspects.

Have you tried doing an update on the phone to see if a restore makes any difference? You can try the "Reset All Settings" thing too; it's always worth trying the software solutions first before throwing hardware at it.

So my suggestion would be to check over the home button flex carefully with a magnifying glass and a bright light. Take a look at the both sides of the connector and see if you can find any bent, broken or missing pins, then check that lightning connector and let us know what you find. After that we can try and figure out where to go next to get your phone back to normal.