Michelle,
Some answers for your questions....
1.) I'd suggest starting the hike in Monterosso (the most northerly village). You can easily take the train from whichever location you're staying. The first leg of the hike from Monterosso to Vernazza is the most challenging, so IMO it's better to get that out of the way when you're well rested. The next leg from Vernazza to Corniglia is almost as difficult.
2.) You should be able to easily hike all of the trails from Monterosso to Riomaggiore in about 5-6 hours. The last section from Manarola to Riomaggiore (the famous via dell'Amore) is mostly paved so it's extremely easy. There's a Bar part way along where you can stop for a "cool one" while admiring the ocean view.
I'm also a solo traveller, but I used a Mar-Mar Rooms in Riomaggiore on my last trip there. The "official" CT Hostel is in Manarola, and it seems to be clean and well run, however they have a strict lockout in the afternoon and that's a problem for me.
3.) I can't recall whether my last visit was 2 or 3 nights? It's not all "couples having dinner" from what I saw. If you're interested in shops, you might want to stay in Monterosso as it's the largest of the villages and has the best beach as well as the largest selection of Hotels, restaurants and other "tourist facilities".
I typically use my time in the CT to do absolutely nothing! It's a place to relax and recover from strenuous travels in other areas. I'd suggest stayinng for 3 days and perhaps take a day trip to Portofino or some other nearby spot. There are some villages above the CT that can be accessed by hiking trails also.
Note that to hike the trails, a permit is required (there WILL be people checking). You might want to have a look at the Italy Guidebook to get some ideas on activities in that area.
Happy travels!