First time in Italy

I've never been to the Cinque Terra, so I'll let the many more experienced people on this site answer the first 2 questions. For the other 2, I'll only point out that you do not have an efficient itinerary as far as train travel is concerned. You will take a train (more likely at least 2) from Monterosso to Pisa, hurry up and do a quick climb up the Leaning Tower, then hustle back to the station to board a train that one hour later, will go through Monterosso where you just came from to get to Milan. In Milan, you want to take a day and travel back to Florence that is just about one hour by train from Pisa.
I think a better plan (one that costs less in train tickets and your own valuable time) is go to Pisa, climb the tower at you leisure, than continue on a regional train to Florence and spend the night. Next day, see the sights that interest you in Florence, and in the late afternoon or early evening, take the Freccia train to Milan.
The Freccia is about the only way between Florence and Milan due to the narrow valley in the Apennine Mountains. It is very busy and only fast trains are operated here to make best use of the tracks. It is THE rail connection between northern and central Italy and carries a lot of traffic.

I have some comments on your first two questions.....

1. There are beaches in both the "old town" and "new town" and I believe they each have both public and private areas. The beach in the new town has more of a sandy composition and is larger than in the old town (although typically more crowded too). Depending on which hotel you stay at, they may have their own section of beach for guests, but if not they will be able to tell you which sections are public. I don't think the cost for a beach chair and umbrella in the private sections is too high, so that might be an option.
2. No, the Cinque Terre Treno card does not include ferry travel.
   

Hopefully the others will have some information on the last two questions.

1. Beaches in Italy are all public. In some cases the State gives a Concession to operators of beach establishments. Such concessions are only up to 5 meters from the water. The first strip of beach 5 mts from the water cannot be occupied by the Concession. The concession also must permit the passage of people to the water without payment of any ticket. Generally the concession will require payment if you rent their facilities (umbrella, chez, changing room etc) and may require you to pay a ticket if you intend to stay within the space occupied by the concession facilities. If you stay within 5 mts from the water, they can't charge you. Nevertheless they will try to ask for payment, sometimes some extortionists will try to get money from you. I go to Monterosso often and They have tried with me and my wife a few times (my wife looks foreign to be sure). All I do is get up, let them know I'm a former Carabiniere and ask them for their Carta d'identita', then proceed to call the Carabinieri of Monterosso (number saved in my cell +39 0187 817524). I also tell them that if I see them asking from money to anybody else within my sight I'll call the Carabinieri. They immediately run the hell out of my face and I don't see them again the rest of the day. I love messing with those low-lifes.
2. No.
3. Doable.
4. Frecciarossa is the fastest way to get from MI to FI. You could also take a series of Regionali with a few changes along the way (Bologna for sure). But they take longer as they travel on the old rail line through the cities of Emilia Romagna. Then they cross the Apennines on the old line through Vernio and Prato. The new high speed line is reserved to the Freccia trains only. I don't know if you refer to the Mall where the famous fashion houses have the outlets. That is not in Florence, it's in the middle of nowhere outside Incisa Valdarno. You need a car or an expensive taxi from Florence because it's at least 25km out of town. There are some companies that organize bus trips there. Never done it myself, since I usually drive to it, when my wife drags me there.

I think you'll need more time in Pisa. There is more to the field of miracles than just the leaning tower! There is also the duomo, the baptistery with a wonderful echo effect, a very interesting graveyard if you are into that sort of thing (the great mathematician Fibonnacci was buried there), and a couple of interesting museums.