All Minecraft Enchantments: The Complete List

43, across current Java and Bedrock Edition releases. A 44th, Cleaving, only ever existed in a Java Combat Test snapshot and was never added to the real game.

There's no single "best" — it depends entirely on the item and what you're doing. Mending (free durability from XP) and Unbreaking (extra durability chance) are the closest things to universally good, since they apply to almost everything with no downside. Beyond those two, the right pick is whichever matches your specific goal (Sharpness for general combat, Fortune for mining valuable ores, Protection for survivability, and so on).

Each enchantment's own page lists its exact conflicts. The two largest conflict groups are the melee damage types (Sharpness, Smite, Bane of Arthropods, Breach, Density all mutually exclude each other) and the four armor protection types (Protection, Blast Protection, Fire Protection, Projectile Protection all mutually exclude each other) — pick exactly one from each group per item.

Yes — and separately from that, every anvil combine also gets more expensive the more times that specific item or book has already been combined before (the prior-work penalty), regardless of which enchantment or level is involved. See Enchanting Strategy for exactly how that works, or just use the calculator above to get the real cheapest-order cost for your exact build.