All trains between your preferred destinations are included.

Note that Eurail is not a train company operating trains, but a ticket that allows you to board trains of many different railway companies. There's a list of companies included on the website. 

Local busses, tram, metro's are not included. City transport is usually very cheap, only costing a couple of Euro's (Remember that most European cities don't like cars, so they have to offer people an affordable alternative). 

Timetables are not known yet for that period, they will start to show from October. Note that some countries can be notoriously late in uploading there schedules (like Italy). Don't panic if you don't see them show up on your dates until late the end of this year. This is because the international yearly timetable update late December. Usually doesn't change much with current timetables. 

Use www.bahn.com to plan, or the respective national rail planners. Try a random date next week to get an indication how long it takes to get from A to B. That way you can start calculating how much time you need, but I guess you'll be fine with a 15 days in two months pass or a monthly continuous pass. 

You'll need reservations for the Eurostar from Paris to London, best reserve as soon as you can. It's very popular on holiday periods. Www.raileurope.com is a good place to do this (desktop version only) 

For more information I'lld suggest taking a look in these webpages, explaining you the ins and outs of the European rail system: 

https://www.seat61.com/european-train-travel.htm

https://www.seat61.com/how-to-use-a-eurail-pass.htm

Some great advice from @BrendanDB there.

On your itinerary, I would add the following

Geneva? Unless you have a special reason to go there, I’d skip it. It’s not going to win any awards as a tourist destination. Especially since you’d be going in the wrong direction for Italy.

One day in Rome? On a day trip from Florence? You’re not going to be able to do it justice. 

You also don’t seem to have any time in Florence? Getting there will take half a day, and getting to Paris will take all day.

Can you drop an Italian destination, or add an extra day or two to Italy?

Thank you @BrendanDB for your detailed response, gives me some confidence in planning for my trip now :) 

Also thank you @ralderton, will definitely re-plan my trip around Italy haha.