I have much food and drink, 185 medical supplies, etc.

About 1/2 a day out a crew member shows injuryfor no good reason and morale drops below 50. But how the #(*&%(*# do I heal this crew member? I keep reading about a resting place, but huh? I've tried moving the crew member to literally every spot on the ship's menu and she doesn't go anywhere.

To top it all of IT'S THE BLEEDING DOCTOR WHO IS INJURED!!! Seriously?

Also, I keep losing morale because of water -- but there is 365 water on board! And rum, and beer and other morale-boosting stuff, and food galore, yet I'm losing morale? What on Earth?

Also, if crew members get injured this frequently, my morale will be zero in about three days. What's with that?

Well water has -1 morale. It says so in the description, so that's normal and not a bug. There are other beverages that have 0 effect on morale or even add some. The food/drink works like a queue. It's always the first spot that is used up first. So if you have beverages that add morale, you have to place them in the first slot.

Injured crew members can only be healed in the reserve spot, which is in the bottom right of the ship screen.

I don't think my ship morale ever dropped below 80.

The Kraken pub/starting island has one surgeon for hire, and Nekataka (the second island where the main story continues) also has one. They did for me, I'm not sure if the ship crew members are randomly generated.

If you place an injured crew member in those bottom right resting slots, they will use healing supplies whenever the day rolls over. I think its 5 medical supplies used per day of healing. So you have to move the ship about (or wait?) to tick over the time of day.

I really want to enjoy this ship management game, but whenever I board enemy ships its all for naught, the enemy crew is hale and hearty despite being ripped to shreds with grapeshot and without fail every single enemy crew member who can see me will fire on my main character in the first round. They never target anyone else. I'm not even in the front row of the formation!

No, the ship crew members are not randomly generated.

Healing takes up 5 medical supplies if a surgeon is in their slot. It's 10 without one.

Oh no! The enemy has learned to focus fire! The fiends!
I think it is just bad luck, or maybe the AI targeting the weakest/easiest target for their attacks (this may be the case, if you have a look into the settings of your characters behaviour when it comes to their specials).

I Like ramming and boarding, but I also find it much easier to fight the battle from a distance. Move close in, within range of your guns, move alongside the enemy ship, and fire. Then jibe, wait one turn, and fire again. Rinse and repeat. It helps to upgrade your weapons. This, I find, is by far the easiest way to win battles at sea. I'm using the original ship with upgrades but I always keep the fastest combat speed so as to be able to dance around the heavier enemy ships.

There's definitely some charm to the ship battle, a little too much 'continue' button. You can deck out your ship to be a beast in ship to ship combat very quickly, even the smaller vessels. But you get less tangible loot this way.

Boarding is more profitable and if you recruit team members (not ships crew) all over Neketaka, and take them out occasionally just to work through their level ups. Every time you board a ship they will be there on deck with you, running their ai routines and effortlessly slaughtering the crew of most any ship of-or-near your player level. Even better if you give them a few appropriate fine/exceptional/superb gears rather than just selling it all to the next vendor you find.

You will most likely be face planted in the first round without an anti-death spell or by playing as a tank, but you'll still win effortlessly. Boarding doesn't require much of any ship upgrades but you should still set aside some money for upgrades.

Aye, what PorkDish said, plus what he said in my words:

Boarding:
It can be fun and boarding leads to more loot. Basically you need enough ship health to soak up maybe one salve from the enemy while you close in.
Crew members can fight in the battles, but you are better off with party member (be they the pre-made game ones such as Eder or ones you "roll up" from the inn's).
Make/recruit as many as you can, remember to upgrade their gear with your cast offs, and also remember to invite them into the party sporadically just so you can level them up (all out of party characters gain exp while you do, but at a lesser amount. So they will level up, just slower, than the PC and in party members).
You can only level them up on land (and at an Inn where you can change your party members) which is a tad annoying and fiddly.
The "unity console mod" (it is a good mod I recommend btw) on nexus allows you to switch party members anywhere, but you will still need to be on land to level them.
Pre-made followers will start at the level of the main character at the point they are first recruited, rolled up ones will be from level 1 (minimum) to your level -1? (maximum) and cost you 250 * level to hire.
Crew members have their set AI (classes) and level up (my Chitupec is level 9 or 10 now) through ship to ship combat (and I think sailing into storms as well), but their equipment (unless it scales with their level) is meh (apart from Audr who has decent gear) and cannot be changed (unless you mod it and start a new game or do some fiddly stuff with unity console mod I guess).
I have added their classes (and in the middle of their equipment) to this website (someone else had already put in the rest of the info):

[url=https://pillarsofeternity.gamepedia.com/Pillars_of_Eternity_II:_Deadfire_crew_members]https://pillarsofeternity.gamepedia.com/Pillars_of_Eternity_II:_Deadfire_crew_members[/url]

Forgot this bit about boarding: You can board when you get to a distance that is less than your current combat speed from the enemy.