2002 acura mdx will not start

So no crank, (the engine doesn't turn) and no start?

Did you disconnect the battery during the starter install? Did you reset the anti theft immobilizer afterwards?

https://www.google.com/search?q=2003+Acura+MDX+no+crank+no+start

..

ANY CDES?

you more then likely have an anti theft device, trying putting a flasher on either left or right and try, due all the dash gauges light up when you put the key to the on position? you can try unplugging battery and replugging it, ensure you have the radio code for stock radio before doing so.

Start with the first mechanic who replaced the battery and have them check the charging system. (Alternator) They should have done this when changing a battery, to rule that out as a problem in the first place. Explain the problems you have been having, as a bad charging system can cause all sorts of electronics problems and have them check the engine codes and have them explain what they are.
You could also go back to the muffler shop and tell them their sensor replacement did not fix the problem and have them correct their work.
Diagnosing Alternator Problems EricTheCarGuy

Check ignition switch plug for any sign of burning

If you thought it was the battery, I'm guessing that what you mean is sometimes it will spin the engine and start, while other times it will not spin the engine at all. If this is the problem, I would suspect the starter solenoid, starter or starter relay is failing. The solenoid is most likely.

corroded battery terminals, clean them, and try jumper cables.
both way use same crappy terms.
what transmission A/T or M/T my guess, A/T
so wiggle the gear shift, try neutral starts. this is the PRNDL switch
wiggling it finds new fresh brass on the switch contacts,
wiggle as you crank the key.
if that fails
then hot wire the starter solenoid.
if that act causes cranking, then the starter and battery and cables are good. and means the KEY LINE is bad.
if no crank , hot wired, then that means the starter bad. (battery good , battery charged in all cases)

see hot wiring here.

http://www.fixkick.com/nocrank.html

in a nut shell
1: battery charged at 12.6vdc. rested.
2: chock wheels, all brakes set., in PARK (or M/T in neutral.)
3: gain access to the starter
4: unplug the key line (that is the tiny wire) on starter solenoid.
5: keys in pocket
6: jumper the solenoid tiny wire, to the big outboard lug on solenoid.
grrrrrrrrrrrrrrrrrrrrgggggggggggggggggggggrrrrrrrrrrrrrrrrrr
it wont start, keys are in pocket.
the starter is good, i save you 1/2 the diagnosis. in 1 simple step
put the wire back removed in #4
if its dead, the KEY line is dead. ask how to fix that. next. i f so be...

Transmission range switch ?

SCAN VECHICLE CHECK FOR TORQUE CONVERTER FAULT OR TRANSMISSION FLUID PRESSURE SOIENOID. CODE.AND CHECK TRANSMISSION FLUID LEVEL