Boost HP-draining damage moves such as Giga Drain? If so, will it affect non Grass-type moves such as Drain Punch?
Well sorta. You gain more HP back than normal (30% more).
Boost the amount of HP recovered from moves such as Leech Seed? If so, does Leech Seed deal more damage?
Yes if does and no Leech Seed doesn't deal more damage instead you recover 30% more than your supposed to.
Boost the amount of HP recovered from moves such as Aqua Ring or Ingrain?
Yes it will .
Boost the amount of HP recovered from immediate healing moves such as Synthesis? If so, will it affect non Grass-type moves such as Roost?
Boost the amount of HP recovered from non-immediate healing moves such as Wish?
In any case the Big Root lets you recover more HP from HP stealing attacks such as Drain Punch and Leech Seed. It recovers more HP by 30% but does not increase the damage of the attack.
Here is the Smogon and Serebii description:
When this item is held by a Pokémon, if it uses the moves Absorb, Aqua Ring, Drain Punch, Draining Kiss, Dream Eater, Giga Drain, Horn Leech, Ingrain, Leech Life, Leech Seed, Mega Drain, Oblivion Wing or Parabolic Charge, the HP it recovers is increased by 30%. ~ Serebii
If the holder uses a draining move (Absorb, Drain Punch, Dream Eater, Giga Drain, Leech Life, or Mega Drain), the HP gained is increased by 30%. The HP gained from Ingrain and Aqua Ring is also boosted by 30%, bringing them to 8.125% max HP healing per turn. Leech Seed heals 30% more, but doesn't inflict more damage on the target. It does not increase the health gained from Heal Order, Milk Drink, Recover, Roost, Slack Off, Softboiled, or Wish. Affects Liquid Ooze. ~ Smogon