Changed exhaust did not rejet pops when ideling

Get the fueling set up on a rolling road. Honda set the fuelling to suit the Emissions and the Exhaust they fitted.An after market exhaust will usually have less backpressure and need a set up. When you say backfire, do you mean a great big bang or just a little popping sound from the exhaust? If it is the latter, it isn't a massive problem but a set up will cure it. A Big Bang could be a cause for concern as it means the ignition is taking place when the Exhaust valve is open and firing in the exhaust itself. This will give a drop in performance and may cause the engine and exhaust to fail.

Intake leaks, broke/open vacuum lines, or carbs that need cleaning correctly will and can cause a pop from the exhaust

Backfiring out of the exhaust is usually an ignition problem. Unburned fuel is getting into the exhaust system because it's not being ignited in the cylinder. When the engine fires on that cylinder again, it ignites the raw fuel in the exhaust causing the pop that people call a backfire. I'd investigate the ignition system. Connect a timing light to each of the spark plug wires if it has a single fire ignition or either of them if the ignition system is the stock dual fire. Start the engine and watch the timing light to tell if the engine is missing as the light will miss a flash.

Good Luck
Steve

If the mechanic did not replace the exhaust pipe gaskets in the head or on any connections between the head pipes and the mufflers , the air leaks could be igniting unburned fuel in the exhaust , causing the popping and back firing. Try replacing any gaskets in the exhaust system.
If you have a flat spot or hesitation when accelerating than you probably also need to rejet the carburetors. Bigger main jets should fix that condition. A jet kit may also contain differently tapered needles for the slides to help compensate for the change in the acceleration curve.

Popping during deceleration is unburned fuel igniting in the exhaust system. Four stroke dirt bikes are highly prone to this. However, you should be able to minimize the popping by restricting the air leaking into the exhaust. Replace the exhaust gasket in the head between the head and head pipe and the head pipe to the muffler. If there is no gasket between headpipe and muffler you may be able to improvise with a strip of fiberglass cloth

carb cleaning needed or could be vacuum issues

you might want to replace the plug end. I had a 99 neon with the same ploblem changed the sensor but the plug was shorted, changed that and it run great ever since.

not nessesarily.
changing the exhaust system usually requires rejetting the carb.
a different exhaust pipe will give/make the exhaust back pressure change. you should of gotten a paper with instructions saying something about changing the jetting.