JASON G. - Welcome to the forum. Like Frank says, engines need a richer (More fuel) fuel mixture to start, especially when cold so the carb has a choke to allow the vacuum the engine produces to suck in more fuel when starting. In really cold temps the engines even need to run for a few minutes with the choke partially closed to richen the mixture till the engine warms up. It's all perfectly normal.
On Your PTO clutch, sounds like the three adjustment screws on the clutch itself are set properly, but the rod that runs from the PTO lever up to the lever at the front of the engine that pushes in on the center thrust button to disengage the PTO could use adjustment. There's a turnbuckle and a locknut in the middle of the rod, all 1/4" coarse thread, a 7/16" open end wrench should loosen the locknut and a 1/2" or 9/16" open end wrench should adjust the turnbuckle. You want to SHORTEN the rod maybe a quarter to 1/2 turn so Your PTO disengages when You pull the lever. I think the IH operator's manual for those clutches says a "Matchbook Cover" (or any other piece of cardboard like a cereal box or soda 12-pac box) should slide between the center button of the clutch and the fiber thrust button on the lever but You can actually run them just a bit tighter than that until the new friction disc wears in.
I'd "CAREFULLY" run the engine & mower while watching that You don't get toes, feet, or fingers into the mower housing or any other moving part and make the adjustments to the adjusting rod so the mower blades stop when You disengage the mower then tighten the locknut.
I had a CC 129 that had that same problem and I actually had to pull the lever back further than the normal latch allowed to get the blades to stop but once they started slowing down they would stop and I could let the lever return to it's normal "OFF" position. Not really a problem for a careful operator and something quickly fixed with the simple adjustment described above.
Don't know what to say about Your engine bogging down when going over rough ground. Only advice I can offer maybe is SLOW DOWN! The descriptions "Rough" and "FAST" are very subjective especially over the Internet. Sounds to Me like gas in the carb or fuel tank may be sloshing around enough the engine is actually running out of gas or flooding with gas. Also could be a loose electrical connection or bad wire in the ignition system loosing contact shutting the engine off for a second.