Regarding the travel to Rome, the quickest way is the Leonardo express (non stop to Termini train station - ~centre of Rome and quick access to the metro). It takes around 30 minutes and costs, if I'm not mistaken, 14€.
There is also a cheaper train that takes you to Ostiense (quick metro access) for 8€, but it takes longer as it stops at every station in between.

Finally, you have the chance to take the Alitalia bus but I don't see the reason why you should, also if you land during rush hour it will take a lot to get to the city.

Taxis have a fixed price of 48€, but the rush hour issue still applies.

I can't comment on the luggage as I almost never check the bags.

Have a great time in Rome and if you can try to get off the beaten path for lunch if you want to taste some great food; or at the very least avoid like the plague those 'restaurants' that have big menus with pictures of the food on it, these are generally just scam for tourists and you can experience far better things just round the corner (sorta).