So you bought it from a dealer? give you any warranty??
You say "pump not a prop" - so you mean it's a jet drive (no propeller - shoots a water jet out of a (nozzle) like a sea doo?)
Yes, 2 strokes do smoke a bit on start-up especially if they have been sitting more than a couple of days. The fuel in the system evaporates a little leaving the oil behind. When you fire her up you will get some smoke. It should be blue-ish in colour and should go away after a few seconds, once the engine warms up a touch - unless it is running waaaaay oil rich.
And yes, it may take a little for the motor to start peeing, but it will do it at idle and it should be quite strong. If you hold your hand under it at "anytime" it should be nothing more than warm - nothing warmer than "bath water" for sure.
The exhaust relief (a hole or holes at the back just below the pee stream and on the leg of the motor itself) should always be spitting water even if the thermostat(s) have not opened yet.
And yes, at higher rpms the thermostats are bypassed whether or not it's up to operating temperature and you should get a strong pee stream.
The motor should never be too hot to touch. The operating temp is in the 140-165 degree range which is quite warm but should not burn your skin or anything like that.
You should never get steam out of either the tell tale or the exhaust relief - if you do that is certainly an indicator of a cooling system issue.
Now, you are not sure about the status of the alarm. Did it give you a short beep when you first turn the key?? It should to let you know it's working. That would at least confirm the alarm module is present/connected but not necessarily that the temp sensor is connected.
Aside from it sounding like it overheated that "squeak" is more than a little concerning - that's the sound of a fried powerhead - keep your fingers crossed.
This motor would have been oil injected from the factory (on-board oil tank). Was/is it still functioning? Did you mix your gas with oil or did you rely on the oiler?
At this point have you done anything with the motor - tried to start her up?
If not I would suggest you pull out the sparkplugs and try and turn the flywheel by hand - see if she is free moving or you have something welded in a cylinder.
If it moves freely I would do another compression test - a regular automotive gauge from a discount auto place will work fine. You are not so much looking for an absolute number (like 115 psi), but you want to make sure all cylinders are within 10-15 psi of each other.
Depending on what you find then you can move on to the next steps of figuring out what happened...
