Well, on the bright side you have started and run her!! Going to break your questions down and give some thoughts or suggestions.
*** Before you try to adjust your carburetor and adjust running. Replace the cylinder head and check compression. Please clean the mating surfaces well and use a new cylinder head gasket. No need to add permatex or whatever sealant it will be fine just as is.
Your manual will have the procedure how to clean your cylinder head mating surface and checking for true flatness. Section 5 page 9. This is very important for a good seal.
Look in section 5 page 13 and look at diagram 5-33 for your cylinder head screws torque pattern. Your torque value will be 96 to 112 inch pounds. Please note I said inch pounds.
First round of torque use 1/2 torque value and follow the pattern, the second torque pass use 100% of the torque value and follow the torque pattern. After you run the motor for a while and she is good a warmed up, the then re-torque a third at 100% torque value following the torque pattern.
You may have to remove the thermostat cover check the thermostat, to see if it is good. If the gasket is damage, will have to replace the gasket so it does not leak....(I know more supplies...sorry).
You can test your thermostat in a pot of hot water. Place the water temp at about 160 degree F and place the thermostat in the pot and in a minute or so you should see the thermostat open. 8)
Running with the choke out.
** Usually when someone states that the motor is only running with the choke out, it usually means a lean condition carburetor wise. (Not enough fuel get to the motor and pulling the choke out creates a richer fuel delivery.
** Please make sure you have fresh fuel oil mix 50:1. Old gas mixtures of greater than 3 or 4 month can accumulate water if is no stabilizer in it.
-- Did you prime up your fuel system by pumping the fuel bulb until it was somewhat firm?
-- Is the fuel hose connected to the tank with bulb connected closest to tank. There should be a little arrow pointing out the direction of the fuel flow to the motor. (Make sure the fuel tank vent cap is open)
-- You may have to have keep the choke out for a little bit and then push it in when it is warmed up. (Does it kill when you push in the choke or just run a bit rough?? If it runs rough and you pump the fuel bulb and it perks up you make have a fuel pump issue)
Your initial carburetor settings:
-- Make sure you turn your slow idle needle in (clock wise) completely and gently seat. Then turn it out (counter clockwise 1 1/2 turns out) Take a dab of paint or whatever and mark the knob end of slow idle needle, so it can help better determine if your turned out 1 1/2 turns. Once you have the motor warmed up and choke pushed in. Try to turn the tiller throttle to lowest position to keep the motor running. Start to turn the slow idle needle in 1/8 to 1/4 increments clock wise (wait about 10 to 15 sec in between turns to allow time for the motor to react). The motor should start to rev up as you adjust the slow idle needle. You then lower the tiller setting to a slower setting and continue turning in (clockwise) the slow idle needle until you get a lean sneeze condition (like a misfire or missing jerk). You then turn it out about 1/4 out (counter clockwise). This should be the smoothest running condition at this setting and you should not have mess with it unless you have rebuilt your carb or another cleaning.
**** Because of the low profile of this motor, it sits low in a barrel and it prone to inhale its own exhaust and runs rough and makes difficult to adjust idle in the barrel. To help with this place a fan blowing on the motor to prevent the accumulation of exhaust gases. This is not an issue at the lake because the motor is in the open and exhaust will flow away and prevent recirculation. Also you will be moving along as well. You will make your final slow idle adjustment at the lake under load with the motor in forward gear to get your smoothest slow idle condition.
-- If you are still have issues with fuel delivery and still have to keep the choke out, you may have to
1 Revisit your carburetor and re-clean it. ( Sometimes there is something that was missed)
2 Evaluate your fuel hose and bulb system too make sure it does not have cracks or compromised to allow air to enter the system and displace fuel and cause a lean condition. This includes the internal fuel hoses in the motor as well.
3. Evaluate your fuel pump.
Your motor not pumping water.
-- 1st make sure you have that lower unit in a barrel or can that can allow you to raise the water level up at least half way up the mid section and at least 6 inches over the section where the water pump housing is located.
The water will come out as a mixture of exhaust gases and thick spray out of the back where the exhaust relief port is located. That is your water indicator.
https://www.youtube.com/watch?v=xPwfWgNRo_M&feature=related
-- When you replaced your impeller, did you make sure the impeller pin or key was in place. If not then there may not be anything to drive the impeller to turn along with the drive shaft. (We have all done that...lol)
-- When you lined up the water tube and the water pump housing gromit it could have come out or misaligned while trying to adjust get the lower units and the exhaust housing to mate up. So either the water tube is not in the water pump housing gromit and not sending water up to the head or the water tube could have push down on the water pump housing gromit and cause it to distort the gromit. This will cause an obstruction. Lube with rubber gromit with a little oil allow it slide in easy. Pay close attention when lining up the lower and mating the two halves, so that you notice the water tube easily slides in the gromit.
-- There is a water tube gromit located at the base of the motor where it connects to the exhaust housing. If that gromit has become damage or distorted or the previous owner accident pulled down the water tube and tried to shove it back up and it did not communicate well then you would have a obstruction. (The only way to resolve that issue is to remove the outer motor housing and remove the power head from the exhausting house base to evaluate the tube and gromit connection or lack of. (This would take a good afternoon and a 6 pack of beer....but I've done it before....lol :LOL2: And lots of pictures.)
-- When you remove your cylinder head cover, this would be perfect time to evaluate your water jackets and passages to make sure they are not filled or obstructed with sand, silt, salt or junk to prevent water flow and prevent good water circulation and cooling. Please protect the inside cylinder wall and chambers so sand, grit or what does not get in there to later cause damage/scoring.)
Here is a you tube video of a person using a drill to turn the drive shaft and test to see if the water pump is pump water out the gromet.
https://www.youtube.com/watch?v=KIt1HTL096w
(Make sure the motor is in neural and protect the end of the drive shaft with some like rubber than attach the drill connect or ratchet nut and turn it.) It should shoot it up good. This will rule out your water pump and allow to concentrate on possible obstruction mentioned above.
I will post addendums if I think of something later on.
I hope this helps you along.