Use your character's backgrounds to build intrinsic motivation for your players' characters.
If you're looking at the LMoP adventure, you hopefully have the pre-built characters handy. If you look at their backgrounds, you can see that for every character, there is some kind of intrinsic motivation built into their backgrounds for at least one of the elements of the adventure.
This provides the characters with clear motivation for why they should care about both the mission and the results of the mission.
You can do this in a custom setting in a couple of different ways. First, you could assign different background elements to your players, this may be a bit of a deal breaker for some players, but I've had good luck with it using a pre-built adventure and giving my players the option of which of 10 or so to select (Horde of the Dragon Queen provides 10 different optional background elements).
However, the best thing you can do is to pick up plot elements from your characters' backgrounds. Have them write a short paragraph or even just be super detailed when they are building their BIFT part of their background so that you can mine that for details and NPCs and other motivations.
If you're looking to provide intrinsic motivation, having it worked into your characters' backgrounds is the best and easiest way to do it.
How should I go about choosing an adventure for a group of new players?
The most important thing is to find something that speaks to you. If the premise leaves you flat, you are not so likely to be able to breathe life into the campaign.
The other most important thing is to find something your players will enjoy. Heck, they'll probably enjoy anything you DM cuz you'll be so great, but if they are all obsessed with dragons, maybe the module with the dragons, right?
What qualities should I look for?
Make sure the module you select is for new first level characters.
(Unless you really want to start at a higher level, which would make
your work a bit harder.)
Modules labeled introductory are good for starting DM's and players.
They often reprise the rules you will need to be familiar with, and
give points to where to look rules, etc.
CAVEAT: The word "introductory" doesn't have a fixed definition. Some introductory modules will be entirely self-contained, while others might require other materials, such as the Monster Manual and the DM Guide.
A module might be (A) specifically for particular game rules (like
D&D 5e), (B) be "compatible" with a set of game rules, or (C) it
might be just the "story" part without the monster stats. A module
of the first type will be easiest to use.
...and how can I tell if a published adventure has them?
I think that a trip to your Friendly Neighborhood Game Store might serve you well. The staff of typically pretty knowledgeable, and you can flip through the merchandise. Other customers often might offer their insights as well.
This site and others have chat forums where you can get suggestions and discuss.
You might also read product reviews online. For the in-depth info you are interested in, I'd favor review website like Escapist Magazine over reviews at online stores (although both have their place).
Should I restrict myself to official WotC adventures on the presumption that they have the best writing and playtesting or is there a way to identify third-party adventures with a similar (or higher) level of polish and quality?
Probably not a bad idea to favor the WotC's D&D 5E material, if that's what you're going to be playing. This has more to do with the encounters being tailor-made for 5E, than one company's material being better than another.
So which are those?
Applying my own answers to your criteria, The Lost Mine of Phandelver in the D&D Starter Kit, and Tyranny of Dragons both would be solid choices. Phandelver also would be a good fit for the number of play sessions you mention.
Tyranny of Dragons is an updated version of the early D&D 5e releases: Horde of the Dragon Queen and its sequel, Rise of Tiamat. Its updates include play balance fixes, perhaps most notably in earliest set of encounters, which make this module more appropriate for beginning DM’s and players.
NOTE: More recently, another official introductory module has been released: Dragon of Icespire Peak. I haven’t read through this one yet, so I cannot comment on its specifics.
You'd mentioned social encounters...
Most D&D modules have a whole lot of fighting involved, and the two I mentioned are no exceptions. Without doing any spoilers, Phandelver probably is more dense with opportunity for social encounters than Horde of the Dragon Queen.
The existence of opportunities for social encounters, of course, has a lot to do with your DM'ing. Monsters don't have to attack the party on sight, even if the module says they do.
Best Answer
There are not really any official free adventures yet. EN world had a contest, sponsored by WoTC for short adventures for the D&D Next Playtest rules. You can get the adventures here (there are 8 of them) While I have not played them, the winner of the contest, "Mind Blast" has gotten some good reviews. Another award winning module is the Wizard's Amulet by Froggod Games.
However, in 2015 WoTC plans on revealing plans for 3rd party support and many are speculating a type of marketplace for user content. Presumably you will be able to get free adventures from that system. The stated reason for this being opened up only in 2015 is so that people have a few months of playing the game before they start making content.
Ofcourse, there will be lots of people playing games on roll20 or G+ with the purchased modules, and there are broadcasts of the Starter Set being played by WoTC employees.
Two more free 5e adventures were announced today, July 6th but their quality is untested.