Which town would you recommend staying in of the five?
Opinions will be all over the board - and none of them will be wrong - but we liked Monterosso because it was the largest of small villages, and had the most visitor amenities. It's also a good spot to access the (usually) the northernmost leg of the Sentiero Azzurro (Blue tRail), which leads to a next question....
Is the Blue trail fully open currently?
No and it will not be fully open this season. That said, the CT has MANY trails so familiarize yourself with the Parco Cinque Terre's trail system. This is the website, if you haven't found it yet:
http://www.parconazionale5terre.it/Eindex.php
This page will show you what trails are open or closed (black lines) at any given time. The "Blue" encompasses the four SVA2 segments, and the Monterosso> Vernazza and Vernazza > Corniglia sections are usually open. Corniglia> Manarola and Manarola> Riomaggiore have been closed for some time and will remain so this year. Last I saw, reopening for those two legs are scheduled for 2024 but I wouldn't hold breath on it. As well, open trails can close to weather, rock slides or unexpected damage at any time. Anyway, you will need to purchase a park pass to hike any section of the Blue.
http://www.parconazionale5terre.it/Ecinque-terre-card.php
There are workarounds for Corniglia> Manarola and Manarola> Riomaggiore, albeit longer and more strenuous: spend time exploring the site for all of the park's trekking options. Clicking any line on the map will bring up a description of that trail, its difficulty, estimated time to complete, etc. There are also some suggested itineraries.
Half of me is like “include Amalfi!” And the other half is like “don’t
be crazy, stay close!”
Nope, you don't have time for that one. Other than you might want to stay in the airport city of departure the night before your flight, devoted hikers can find plenty to do in the region unless it rains. As suggested, you could day trip to Porto Venere, Santa Margherita Ligure or Lucca (walk or bike the city walls there). As much as you prefer smaller to larger towns/cities, look at Genoa and Pisa.
What are some “must do” recommendations for a first timer?
Entirely up to you. While we enjoyed some hiking, we also enjoyed exploring the local churches and architecture in general, shutterbugging about (me), and exploring some of the history. Oh, and I dearly loved Florence but I'm an art geek so... That was a "gotta do" for me, and for far more than just a day. There is no "gotta do" unless it's your "gotta do". :O)