Brian,
The clicking of the solenoid suggests a couple possibilities - Bad battery or improper ground connection - one quick way of eliminating the bad ground is to turn on your headlamps, have somebody watch the lamps and turn the ignition key to start - If the lights go down or out immediately it can possibly be a bad ground - However, the cables getting very hot does suggest either a dead short somewher in the ignition circuit, improper connections, or bad starter - If the store where you bought the starter offers free testing (Like Advance Auto Parts, Pep Boys, etc.) I'd pull it off and have it tested - If It's bad , or weak it'll show up on the bench and you can apply for a refund or swap it out. When I buy parts from these jokers like starters or alternators I always go for the one with lifetime warranty - If you keep the car forever you'll never have to by that component again - just bring it in for replacement when it fails.
Last idea - Is the starter pinion gear jammed in the ring gear teeth? - Occasionally the starter in my Sprite will hang and produce much the same symptoms as you describe - I give the starter a moderately enthusiastic rap with a ball peen hammer and that usually frees it, or put the car in gear and push it forward and rock it to and fro - when the pinion gear releases you'll hear a distinct "PING" and you know it's free. - If the condition seems to be chronic you may look into whether you need to shim the starter to get the correct engagement ( Were there any shims under the old starter when you dissassembled it?)
Anyway, Your question will draw a bunch of good replys from the forum members, so don't despair -
Good Luck!
Bob M.