Here are a few ideas:

Is the float level set correctly?

Does the throttle roller piece touch the ignition plate extension (the part the roller can roll against) where there is a marking (a little line) on the ignition plate extension?  I’m not certain your motor has this marking, but if it does, that should be where the roller starts/stops contacting the ignition plate extension.

I would double check the carburetor is indeed clean.  Debris from other parts of the fuel system could have made it to the carburetor.  The likely suspect area is the low speed circuit.

For the water pump, the water should come out an approximately 1/2″ hole around halfway in between the powerhead and the gearcase.  Unless someone drilled another outlet, it should only exit from there, so not sure what you mean by “both sides”

Another thing is that you should give the motor a gear oil change as well.  The previous oil may be dirty, contain water, or it could have been drained.

You have alot of questions, will answer a few of them.   First, like others have said, that water level in that bucket should be up over the gearcase/exhaust housing joint so the pump can work properly.  You don’t want to ruin the new impeller you just installed.

Your next picture shows where you will normally see some water dumping out under those lower mounts, this is normal and expected.  Hard to explain, but that water helps seal the top of the pump and needs a place to escape…. The exhaust cover on the powerhead should always be relatively cool.  The engine does have a thermostat, so the cylinder head should get warmer than the exhaust cover.  Hard to know if yours is getting too hot though.  The thermostat is hidden under that little triangular cover and is easily accessed, the passages might be plugged up as well.  Be sure to order a new thermostat gasket before taking the cover off, those gaskets are unique, and might take some time to be found.

OK, the engine won’t idle, but popping the choke quickly seems to help temporarily, is this correct?  If so, first know that the 10hp carb linkage is tricky, might be binding, or perhaps the choke linkage is messed up.  Any of these conditions may be holding the carb butterfly open slightly while at idle.  Your picture clearly shows the roller is not touching the mag plate cam, so it doesn’t seem as if the cam is holding the carb open.  The butterfly must be closed completely at idle, or you will have a lean condition, the engine will probably sneeze/cough/stall out.  Don’t assume the throttle plate is closing completely/not binding, pull the carb off and have a look inside at the butterfly.

Another problem area on these carbs is the pesky threaded “sleeve” that the low speed needle threads into:

item 35

https://www.marineengine.com/parts/johnson-evinrude-parts.php?year=1962&hp=10&model=QD-23&manufacturer=Johnson&section=CARBURETOR+GROUP

The needle valve threads into this sleeve, which is not locked in place…Holding the sleeve in place relies on the packing washers being relatively secured in place tightening the packing nut.  Oftentimes, you attempt to adjust that low speed needle, turning it either way, but actually doing nothing.  The sleeve is just turning inside the carb body, the needle positioning is not being changed.  Don’t ask me why engineering felt this sleeve was a good idea, perhaps money was saved, who knows.

Finally one of your pictures shows a shiny core plug covering that top idle mixing port, did you install that?  Did you use an OEM carb kit, or aftermarket?  Much of the aftermarket stuff isn’t worth bothering with, just creates more problems and frustration.  In any event, that core plug looks like it was just dropped in place without tapping it down slightly to expand/seal its OD to the carb body.  An air leak in this area will create a lean condition at idle as well.

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Sam M.