Couple of notes. I flew from Denver into Munich, via rental car to Italy.
Bought a bike box (aehm, “SCICON AEROTECH EVOLUTION X TSA BIKE CASE”, probikekit.com … there is some money to be saved). I do own a bike bag as well (right, the Scicon AcroComfort whatever). Turns out that some airlines only keep their promised insurance if the bike is in a hardshell.
Gear to take. I got a light weight travel pump (“LEZYNE CNC TRAVEL FLOOR DRIVE”), a leight weight torque wrench (“PRO BIKE TOOL Adjustable T Torque Wrench”), a set of allen keys (staddle, stem, pedals, deraileur), and a flathead/phillips screw driver. As spare only the deraileur hanger (which is impossible to find at a random bike store). For a bike box you have to deflate the tires, take off the pedals and seat, and losen up the handlebar. So there is a minimum set of tools required. Tricky part is to keep the bike box at or under 50lbs (or 23kg), hence the “light weight” above. Of course tubes, spare tires, carbon paste, lubricant, chain wax. Theme for me is to not depend on a bike shop to do the basic things.
In terms of clothing, 2 full kits (shoes, socks, bibs, base layer, jersey, gloves). Chances are you’ll get rained out once, and then not having to wait an extra day for everything to dry up is good.
Arm warmers (2 sets perhaps), gilet (lightweight vest), and/or a thermal vest plus a lightweight water proof jacket, and a more thermal jacket, full finger gloves (water proof), perhaps also full thermal gloves. Weather changes a lot in the mountains, and the descents get cold quickly. I am a big fan of shakedry jackets combined with a thermal vest. For the Haute Route Dolomites I did end up using all of the upper selection … but I guess all of that is personal. The weather forecast was miserable, so in addition to the normal kit (base layer and jersey), also had a goretex infinium jersey with me. Good for a longer ride in a drizzle without overheating.
Safety. Italy requires now a rear light. Got a Lezyne Strip Drive Rear, a tad heavy, but never let me down on 9 or 10 hour rides. Not sure about the front light requirements. Got a computer mount that also has the gopro mount, so easy to add front light. Something to consider.
Logistics wise, ended up with the bike box, and all the rest of the cycling stuff in a carry on. Easily doable.
Bike rentals. I have zero idea. Tried to rent in the US before, and always ended up with ill fitting crap (wrong size, too long stem without spares to change). Friend of mine had rented over in France (for the Alpe and for Ventoux), and while getting a quality bike, on each occasion the frame size was wrong (yes, they pre-ordered, and yes they supplied the measurements, and yes they asked for the proper fame size). If you end up renting a bike, take your pedals and take your saddle with you.