Here's a pdf of some specs
http://www.delavanagpumps.com/powerflo/5800.htm. I believe 16 amps is the start-up draw. My retailer swears they're great pumps but need specific installation. He suggested 12-3 wire because the wire strands are fine making better connection, crimped connections- no wire nuts, short wire runs to the battery, biggest marine battery possible fully charged.
The spraying nozzles cannot be too small or it will cycle like you stated. The pumps will always run smooth at ground level. But I believe once the hoses reach the 2nd or 3rd floor of a roof, the fluid in the hose is creating a head pressure which pushes back to the pump and cause cycling. So it a fine line of using a hose big enough as to not create restrictions and cause cycling, but also not to large as to hold more fluid and create more head pressure.
My batteries are 600 ca or higher and are new to 2years old. They hold good charges from 12.2 to 12.8 volts. The only thing I havent check is the voltage drop when there's a load on.