It kills the engine if you drop the outdrive all the way down? There is no reason for that to happen. Only thing I can see is if you have loose electrical connections, and you are getting a voltage drop when the trim hits the lower limit and the pump draws extra power to push against the pressure relief valve. I would check all the connections (positive and ground) at the battery, starter/relay, ground to engine block, and connections at the ignition coil.
Concerning the hard starting, maybe it was moisture in the distributor cap after not being used for 2 months. Or again, maybe a loose connection at the coil.
A Carburetor that sits too long tends to get gummed up, and run lean, not rich. So that doesn't sound like your problem.
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"Knot Easy" 2000 Horizon 240 Volvo 5.7GS /SX
tow: 2017 Honda PILOT EXL-AWD
prev. boats:
'87 Chaparral 198CXL 4.3 OMC Cobra
'69 Jetstar 16ft Ski Boat, 115hp Yamaha
'68 Aluminum Jon Boat, 3hp Sears
'64 Water Wings