There is lots to do in all those places so it is well worth working out what things and places you are most interested in to determine the amount of time to stay in each place. For example, if you are interested in history you might want longer in Berlin, if you want to see more art and churches you might want more time in Italy etc. Another destination that you could perhaps combine with your trip to Naples would be Pompei - that is a sight definitely worth seeing! As far as saving time and money it could be worth looking at different ways of travelling around Europe rather than just rail - e.g. planes, lift sharing ("mitfahrgelegenheit" in German - a popular and cheap way of travel in Germany) etc. Or check for tickets for trains that cover the whole of Europe - interrail tickets or something.

What exactly are you interested in?

At the beginning of October the weather will start worsening and it will get fresh, even cold. It might be unpleasant to travel across Germany, Austria or the Czech republic. Better heading further south.

It sounds like you already have your plane tickets. If not, look into open jaw tickets (multi-city on airline web sites). Say into Berlin and out of Rome or Naples. Such a ticket should cost about the same as round-trip and save you the time and expense of backtracking. If you already have tickets, open jaw might even be worth the change fee.

When you say Naples do you mean the city or do you mean the area like the Amalfi Coast/Capri/Pompeii?

Also lay out your itinerary in nights, so you get a better feel for how it will flow. For example

Sept 28 - Rome; train to Florence
Sept 29 - Florence
Sept 30 - Florence
Oct 1 - Take train to Cinque Terre
Oct 2 - Cinque Terre
Oct 3 - Take train to Venice
Oct 4 - Venice

and so on.

Look at bahn.de for train times. Here's a rail map for Europe: www.raileurope.com/europe-travel-guide/?WT.mc_id=CJ.europe_map.affiliates&AID=10392997&PID=2334969

As blacky said, it's hard to say how long to stay in these places without knowing your interests.