So, not to sound obvious, but multiclassing as a Druid will increase your versatility the most. The Druid is a full caster with a nice variety of offensive, defensive, and utility spells. As well as the healing you mention. 1 level of Druid would give you 2 cantrips and access to all 1st level Druid spells. 2 would give you Wild Shape, after that it's pretty much all just more spells.
Of the 3 Martial Archetypes, Champion doesn't increase versatility at all. It only makes you better at killing things.
Battle Master only makes you better at combat, but you do get access to a wide range of combat options. It would definitely increase your versatility within combat, just not outside of it.
Eldritch Knight is a bit more interesting - it slowly gives you spells. For the most part, they're restricted to Evocation and Abjuration, meaning that once again it mostly just increases your combat versatility. With that said, some Abjuration spells are useful for things other than killing, and careful selection of the few spells you get to choose freely can massively increase your out-of-combat utility.
So, the simplest option to increase your versatility would be to stick with Fighter and take Eldritch Knight. This could be done in conjunction with a single level of Druid to get all the 1st level Druid spells. The multiclassing rules on spell slots would work out in your favour here; for example, you could cast Cure Wounds out of your highest level spell slots.
If you really want to just boost your versatility, you should dump Fighter and go for Druid. That said, at that point your Fighter levels would be going to waste a bit. If you want to focus on Druid, you should definitely get either 4 or 6 levels of Fighter so that you don't lose an ability score increase.
In the end, you can mix and match Druid and Fighter levels however you want; but you should probably keep them in multiples of 4 so that you don't lose ability score increases. After that, it's just a matter of deciding which Druid or Fighter features you want and taking enough levels to get them.
Yes
"If the attack involves other damage dice" sounds like a pretty solid indicator that "Superiority Dice" would be multiplied, if a maneuver were used that adds the result as damage.
While not applicable to everybody, that sounds like an awesome way to add extra damage to multiply (not that superiority dice are really game-breaking for additional damage) after learning that an attack hit... but I'd allow it, regardless.
Best Answer
The DC for an enemy's saving throw depends on the class feature you have that is causing them to make a saving throw. Generally the DC follows a formula of 8 + your proficiency + an ability modifier.
For any Battlemaster Fighter abilities, there is a section that describes the saving throw (PHB page 73):