Traveling Venice/Florence/Cinque Terre/Rome. Can I hop a train at will between these cities, or will I need a reservation? Mid October, 2 weeks.
Probably if you are pretty flexible with your travel times. If you need to get a specific train you should consider booking. I have only been on one totally full train and that was from Genoa to Milan in the morning - full of business people going up to Milan. I wouldn't worry too much about it and take the chance.
You will need seat reservations for most trains between the cities you mention above. Not a problem. You purchase your ticket and seat reservation at the same time. Buying your tickets 1-2 days before travel should be just fine.
To determine if you'll need reservations, check www.bahn.de (click on the "International Visitors" link at the top). Check each of your rail journeys, and if there's a small red "R" on that line, then a reservation is mandatory. This should also be specified in writing.
On each of the trips you mentioned, there are probably slower IC trains every day, as well as faster ICE trains. The faster journeys generally require reservations.
As mentioned, it's usually fairly easy to buy tickets and reservations a day or two before you plan to travel. I'm assuming that you're not using a Rail Pass? However, even with a Rail Pass, the reservation fees have to be paid separetly (with P-P tickets, this is included in the overall cost of the ticket).