The park's website is also in English; there is a small square up next to the magnifiying glass at the top of the main page for changing language.
http://www.parconazionale5terre.it/Eindex.php
Yes, some of the trails have been closed for quite a number of years now, and remain so this year. You can see which of those they are by referencing this page: a line black in means that trail is closed.
http://www.parconazionale5terre.it/Esentieri-outdoor.php
If the trail system you hiked last time was the very popular 4-trail SVA-2 "Blue Path" (Sentiero Azzurro) between Monterosso and Riomaggiore, 2 of those 4 trails are closed: Corniglia to Manarola, and Manarola to Riomaggiore. Monterosso> Vernazza and Vernazza >Corniglia are currently open. You'll also see on the site that there are many other trails, albeit many of those are longer and more strenuous but the site also provides info on each trail for difficulty, length and est. time it takes to hike, plus what can be seen along the way: just click on any segment on the map and the info about it will come up.
Trails which currently show as open an close abruptly at any time to weather or damage from overuse, or rockfalls. If intending on hiking any section of the "Blue", you will need a park pass:
http://www.parconazionale5terre.it/Ecinque-terre-card.php
Editing to add: So you could hike the SVA-2 from Monterosso>Vernazza>Corniglia and take the train back to your base village. You could also do it in reverse. :O)