New to competitive biking - a few silly questions

 "Is there a specific cadence/RPM I should try to keep steady at for a 100+ mile ride?"

There is no one specific cadence or padallng frequency that is universally optimal. Many theories and philosophies address the question of "best cadence". Most are rooted in lore, a very few in some tangible science.

Optimal cadence occurs when you get the most output (power) from the least input (can be measured by heart rate, an indicator of your physical workload). These parameters can be evaluated using a good bike compouter, a power meter, Computrainer or good old fashioned intuition. If it "feels" right over the long term, it probably is close to optimal. One acceptable generaliztion is that anything below 60 RPM's cadence seems quite slow.

"Is significant soreness in my shoulders normal, and I just need to get used to it, or am I doing something wrong?"

Yes.  

You may be gripping the handlebars a little tight, a pretty common habit among new cyclists. Five-time Tour de France winner Bernard Hinault said you should "hold the handlebars as though you are playing a piano". A moderate, but secure grip is all that is needed. You learn best hand postures. Also, remember that the angle of your handlbars (drops and aerobars) and the position of your brake lever/shifters on your handlebars is adjustable: Find the sweet spot.

Finally, most endurance athletes can benefit from some core and upper body strength; weights, rowing machine, TRX, calisthetics- anything that helps and you will do consistently.

"Should I plan on using a specific hand position in a specific situation?"

Watch the Tour de France on TV and take some tips from how those riders use their handlebars. Always default to the posture that you are most safe and secure in on the bike, but do develop your bike handling skills and confidence in as controlled a setting as possible to improve your skill and confidence.

"After every ride my right leg has a big patch of chain grease on it. Is that normal? Am I doing something wrong?"

Your chain should be clean enough it does not leave a dark residue if touched. That's the ideal, and it is acheivable, but I wager almost no athlete keeps their bike that clean. I've seen top professional triathletes whose bikes were so filthy I would not even photograph them on the bike until it was cleaned since it made their sponsors- and them- look unprofessional.

A clean bike runs better and wears slower. Keep the drivetrain as clean as possible.

Also, if you are using clipless style, ski binding-like pedals, you may simply be touching your chain when you clip out of the pedals. That isn't too unusual, but I think the main thing here is developing good bike maintenance habits to keep your chain clean and your chain lubricant fresh and free of road film, grit and dirt.

I hope that helps. Excellent, excellent questions!