Cinque Terre & Rome

I don't know why you would need a guide to hike on the CT trails, they are more like foot paths. I believe you can hike off the trail but I haven't done that. You'll want to buy a cinque terre card, so I suggest you check out this website (http://www.cinqueterre.eu.com/en/cinque-terre-card) about the card and shows the hiking trails/maps (access to the trails, buses and trains btwn cities)- buy your card at the train station (9am opens) or trail heads. I think you can purchase a multi day card- I don't think you can buy them in advance though. When you go probably just depends on your hiking ability, and how tired you are? There are hikers of all levels. Just bring good shoes and plenty of water! The longest trail btwn. towns is Vernazza to Monterosso and that's a 1.5 to 2hr hike. You can cliff dive in Manarola -in town at the swim spot. I'm guessing the locals cold tell you about other spots. You'll have to check trainitalia for the last train to Rome-- it's a good 3 hour ride from La Spezia (CT to La Spezia Central (~30 min.) and then Rome (3+ hrs)- if you buy those tickets in advance/on line, I believe they are cheaper.

Should I start the hike right away and sleep in the first or second
town the first night (I want to do most hiking as possible! i.e 10km
or so)

Which 'hike'? There are many trails in the park.
http://www.parconazionale5terre.it/Esentieri-outdoor.php
Do note the couple which are closed.

should i sleep in the village before and just start in the morning

Up to you. Much may depend on weather. I highly recommend that you have a reservation for an accommodation as the CT is a mighty busy place.

if option 1 what is a good first stop?

Again, it depends on which trail.

where is there cliff jumping in cinque terre?

I'd ask someone locally. Honestly, there are so many rocks, and the water can be so shallow in many places that I wouldn't do it myself.

should i hire a local tour guide or just do the trail by map?

You don't need a guide. A map is fine if you can follow trail markings.

whats the latest to go to rome on Oct 9 , 2016 - i have a hostel
booked in rome for the night of oct 9

Depends on how late you want to stay, what sort of time you are willing to expend for the journey and how late the desk at your hostel stays open (if it has hours). Explore your options on:

http://www.trenitalia.com/tcom-en

At a quick glance - and choosing Riomaggiore as a random departure village - the very last one I'd take is at 19:25 to Roma Termini: 1 change; total time 3 hours and 34 minutes; arrival in Rome at 22:59. The next (and last) one out of there is at 23:50 and takes 5 hours. 45 minutes so you wouldn't get there until the day after your reservation.