Cinque Terre questions

Total travel time may not be much different, but Option 1 avoids some backtracking. I'd take a "coastal route" train, direct from Rome to La Spezia, instead of connecting through Florence. It's not always very close to the water, but you'll see a bit. It doesn't make much difference which town you stay in. Monterosso is bigger, with more restaurant options and a real swim beach, if you want that, but it's also only 5 minutes by train between each town.

I suggest option 1, as there are a few daily Freccia trains that go from Rome to La Spezia with no changes required. I've taken that train and it works out well. Option 1 also eliminates a very long (about 6 hour) travel day from Cinque Terre to Venice. I used to stay in Vernazza but now prefer Monterosso. There are more restaurant options and nicer beaches. It also doesn't feel as crowded as Vernazza.

Another vote for Option 1 for the reasons already listed. I'm a fan of Monterosso as well as it has more accommodations, restaurants and services, is larger so the crowds have more room to spread out, and it has a hotel we enjoy that is an easy walk from the train station.

I also read that the trails between a couple of villages are closed
due to landslides.

The park has MANY trails but the most heavily traveled are the 4 segments of the Sentiero Azzurro (Blue Path), and yes, two of four have been closed for some years now: between Corniglia and Manarola, and Manarola and Riomaggiore. Last I read, neither are expected be open in 2018. But there are other trails, albeit longer and/or more strenuous. You can also travel between all villages except Corniglia via ferry during the warmer months, and all five of them by train; they are only minutes apart by rail.

For more information, reference the website...
http://www.parconazionale5terre.it/Eindex.php

...and this page specifically for trail info:
http://www.parconazionale5terre.it/Esentieri-outdoor.php

Info on the Cinque Terre trekking and trekking/train passes here:
http://www.parconazionale5terre.it/Ecinque-terre-card.php

No. 1 is more efficient.
With no. 2, from Cinque Terre to Venice you have to pass through Florence (or Milan) and either way it's a long trip.
With no. 1 you break it down to more reasonable travel time.

As stated above, # 1 is the better use of your time. An additional plus is,if Pisa is on your list of places to visit with the kids, you can visit on your way to Florence. There is a left luggage facility at the Pisa station.

Are you planning on using hotels or apartments in Rome Florence and Venice? There are advantages to both which we may be able to help you sort out.

An idea for a tour of Rome here.

http://www.mybesttour.com/prices.htm

We just used them last month, and everyone loved it. If you need more info, PM me.

Option 1 is better.
Monterosso is larger and has a beach if you like that. Vernazza does not,

Thanks so much for the input, everyone. Sounds like option 1 is the clear choice.
Gerri, we are using airbnb rentals exclusively. Works better for a family of seven. I will PM you when I get to the point of booking tours. Thanks for the offer!
Kathy, thank you for the very helpful links. I will be using them!

I forgot one, Jess...
This is the website for ferry schedules and prices:

http://www.navigazionegolfodeipoeti.it/en/

They haven't been posted for next year yet (ferries don't run during the winter) but you can reference it as you get closer to the dates of your trip. I'll note that this is the most expensive way to get around the CT, and especially so with a family of seven so I wouldn't use them as your primary form of transport.

You didn't note the various ages of your family but I'll also note that the trails aren't really cut out for very small children or anyone with mobility challenges.

My wife and I loved Manarola. Small town, some very nice restaurants, interesting hiking possibilities and wonderfully romantic in the evening when the day tourists leave.

I don't know if ending in Venice is crucial for you, but we did this very itinerary this way: Rome-Venice-Florence-Cinque Terre. You can do Rome to Venice as an night train if you want to. Venice-Florence is a direct Intercity type train. Florence-Cinque Terre involves connections in Pisa (the station is 15 minutes from the Leaning Tower -- do the walk) and La Spezia.

I've only stayed in Vernazza and would recommend it highly. We stayed at Gianni Franzi, with a balcony overlooking the Ligurian Sea: Gianni Franzi

Sounds like you have question 1 settled.
As to question 2 I thought Monterosso felt/looks different than the other 4 towns. More of what you might expect to see in a Riveria beach town but less like a small charming colorful rocky coast fishing village.
I did not spend more than a couple of hours there so take my opinion only for what it is.
There are obvious benefits to staying in Monterosso vs. any of the other 4 towns, for some it is the very best option of the 5 towns. If anyone in your party has mobility issues, if beach time is important or even if staying for more than 4 nights I would opt for Monterosso. More to do and see in that town and logistics for day trips is a little better.
My 2 favs were Manarola and Vernazza so would stay in one of those personally. We stayed in Manarola and if returning I would stay in Manarola if a very busy time of year and Vernazza if not as busy a time as Vernazza's biggest flaw is it gets the most crowded of all 5 of the towns.

The closed trails you mention happened many years ago, so anything that mentions the main trail between the 5 towns being open is very old information. Yes there are closed trails but unless a specific circuit within a specific time is of importance for some reason it is not really going to impact you.
Since you have a large party, I imagine some will want to hike, others won't. Those that want to hike will still find plenty of places and trails to keep them occupied and happy. Plenty of hikes starting from either Monterosso or Vernazza, hardest part is usually deciding which ones to do in limited time not lamenting about closed trails.