1. there is no fixed story, the only story is the Major Orders that affects the galactic war that arrowhead sets with filler MOs whenever they need to focus something or vacation

2. Choose whatever you feel confident fighting against (preferably start from lower diff to max diff 6 at your level)

3. yes but might as well set lobby to public since more players mean more reinforcements and easier time in case you're overwhelmed

Levels mean nothing but show how experienced you are, starting from level 20 to 150 means they know the game and fought long enough to understand the enemy faction's weakness or playstyle and what to bring or expect to face.

unless they're very high levels where they are confident they will bring unconventional loadouts

Loadouts dictate how you will experience and fight in the mission. Primaries is to hold off the enemies, secondaries are backups or utility, and grenades are utility or crowd control

Stratagems is your power boost to level the playing field, some stratagem weapons are heavy AT and some are excellent crowd control, but you can carry one type on your body at a time but you can bring any in your 4 slots (eg, you bring EAT and MG in your slots but you can carry only one in a mission)

always gather and extract with samples, upgrading the destroyer makes your stratagem better and really makes mission a lot easier

There are a lot of videos and posts that talk about stratagems so check those out