The Onan engine is not my forte so I'll limit my comments to the generic sort.
Erratic idle might be due to a partially clogged idle jet passage in the carburetor. The "clog" might only be due to some water that has found its way into an idle fuel jet and intermittently blocking the flow of fuel in that jet.
The much larger main fuel jets are not usually affected by a water slug, but the narrower idle jet passes water slugs a lot slower (or not at all sometimes) and can had a profound effect on idle performance.
The prudent thing to do to address that as a first action would be to add some Seafoam treatment to the fuel supply (at the higher concentration ratio) and run the machine through a few tanks of fuel in normal use.

As to the backfiring on shut down:
To eliminate (or minimize) this phenomenon you will need to allow the engine to run at idle for a minute or 2 before you actually stop the engine. Running at low idle for a minute or so will permit the exhaust pipe and muffler to cool down sufficiently to stifle the flash ignition that occurs when the engine finally reaches zero RPM (and all fuel/air flow through the engine stops).

The decision to re-build or replace an engine should be made based on the obvious health of the engine in regard to a number of indicators and tests as follows:
Does the engine consume a lot of oil? Do you observe any visible blue/white smoke in the exhaust stream during normal use after the engine warms to operating temperature?
Do the spark plugs get fouled with a wet/shiny/sticky/gooey tar-like substance?
Does a bona fide compression test reveal a compression reading lower than specification? Does a cylinder leakage test reveal wear or damage to piston rings and/or valves?

As to keeping or tossing the Onan (if major work is needed)
This decision is a matter of personal preference more than an objective summation of facts.
While Onan is certainly a venerable brand (historically speaking) I don't know if you would be well served by trying to acquire a replacement Onan engine.
My personal opinion is that if you must replace the engine you have now........and for $1,000 and more........you would be better off obtaining a Briggs Vanguard or a Kawasaki to drop in there.
Realistically, if you stay with an Onan, you don't have to worry with making custom fit exhaust systems or problems with whether the engine will fit under the hood.

I hope some of the Deere and Onan folks will offer more opinions for you.

drano for Onan and loader build you need join site casecoltingersoll dot com there find where to buy stuff for onan points condenser plugs and wires could be bad. And anytime you get old tractor things like valve adjustment de carbon by removing heads clean carbon off piston and head valve sterns carbon up to. Link below is reasonably on parts Onan. Site above have part numbers service manual for B43 Onan engine in JD 316.

Here is a link that might be useful: parts

My concern is what is going on with the high pitched whine. It may be a simple fix or not.