1) Do all modes unlock stuff?
Yes. Each character has it's own Post-It note with icons representing completion of certain level/boss with this character.
You can check complete list here:
http://bindingofisaacrebirth.gamepedia.com/Completion_MarkDoes playing hard mode unlock more stuff or should I keep playing normal mode for a while?
Regardless if it's normal or hard -- unlockable is the same. The only difference is the colour of the icon -- regular black for normal or black with red highlight for hard completion.
However, if you go for "complete" playthrough you need to do all stuff on Hard with all character plus all the challenges plus some special achievements.
That's quite an undertaking though -- because usually this kind of completion requires several hundred of hours. "Speedruns" from zero to 1 000 001% (latest name for full completion achievement) usually take around 40~ hours and extreme dedication and skill.
2) I haven't made it through greed mode successfully yet. It's that worth doing?
Completing Greed mode with Azazel unlocks Lilith. Overall Greed mode unlocks are not as useful despite this mode being of the hardest things to do in the game.
Overall Azazel has the most useful unlocks across the board: Abaddon, Satanic Bible and The Nail are easily in the Top 5 best BoI items. And the character is kinda supposed to be "easy" character so it's beneficial to do his unlocks as early as possible. Maggy mostly unlocks items that give you hearts and so they're pretty good, Lilith's unlocks are also very good.
Should I be fighting the optional 11th wave? What are the best types of items to prioritize for this mode? It's it best to pick up coins during the fight it leave them until the waves are done?
Greed mode gives you several coins with each new wave, giving you more with eave, starting with 1 then 2 and so on. If you feel you can't survive the next wave you can press the button and stop waves at the expense a half-heart, however coin reward starts from 1 once again from the next wave.
That means it's beneficial not to press the button and kill all the waves in one go.
Unless you have a Jera rune (that doubles everything in a room) there is no point to avoid coins as long as you don't press a button and keep killing waves in time. With Jera you take all 10 wave for maximum coins on the floor and then double that.
The last optional wave doesn't give you coins but if you complete it -- Deal with the Devil room (or Angel Room) opens. DwtD are actually kinda bad in this mode compared to regular modes but they're still worth risking fighting two bosses at once for a possibility of good devil item.
Best items are always Damage Ups and Tears Up -- just like in regular modes. Tears however hit a cap after a while (so taking any more Tears Up items does nothing) but damage can be increased with no limit. Also all items and/or cards with room effect or floor effect are extremely useful in this mode because you can activate a card that gives you stats boost (like Empress, Strength, or Devil) for a room and then stay in the big room killing all waves without leaving while keeping the boost. That's why characters like Judas (whose Book of Belial increases damage) and Eve (Whore of Babylon activates when Eve only has 1 red hearts damage or less and gives speed and damage) are so good in Greed mode because you can stay in the main room for the entire duration of all waves while keeping the boost. Lilith has an item that even stacks with itself each four waves and she's the easiest character in that mode.
3) It's there a point to doing the daily challenge? I've yet to unlock anything from it. Is it just for fun?
There are 2 achievements for doing daily challenges: one for starting 30 daily challenges and one for
completing 5 daily challenges in a row. You can skip days if you see that daily uses a character you don't like, like Lost or Keeper but need to make 5 win streak for second one.
Note that Switch version is prone to crashing after you complete the daily challenge. So after you see the end screen tally of points press B instead of A to prevent that.
4) It's there a benefit to beating the game fast our with a high score?
No, it's just competitive stuff for high-level players and addicts.
However, certain boss challenges and levels only appear if you get to certain boss in less than 20 minutes and to later boss in less than 30 minutes. There is also an achievement for killing one of the very late-game bosses in less than 15 minutes.