Civilization – How does city healing work

This actually sounds pretty subjective. There is only one non-subjective answer I can think of:

Micromanage Early, Automate Late

Early in the game, you have few cities, few population. A couple of well used units can topple another civilization. A worker improving the right square and double your food income. But late in the game, you have lots of cities and units, all the important hexes are improved, you've explored everything already. In short, early in the game every decision affects a greater part of your civilization than the same kind of decision late in the game.

So, micromanage your units early in the game, when they matter a lot. Then, whenever you think they've become unimportant, automate.

Most of the other reasons to automate or not end up being subjective, or simply:

It depends.

Do you want to win absolutely, and play on the highest difficulty possible? Chances are you should be micromanaging everything, checking on every city/unit every turn.

Do you like some parts of the game and not others? Perhaps you like war but not building? Automate whatever you don't like, and play what you do.

Do you enjoy quick games or watching a world evolve with only light input from you? Automate a lot, and click next turn to get to the fun stuff. Also maybe play on quick.

In short, it depends on how you want to play. As long as you're having fun, you're doing it right.

Can you rephrase your question at all to be less subjective, less abut how you specifically might want to play? Maybe, "which of the automation options tend to make good/bad decisions?" We can't tell you how much to care about automation, but we can tell you where you might want to care, if you decide to.

I believe that all ground units that are air-intercept capable can protect in a radius 2 hexes around them. According to research done by Oak, air units can intercept anywhere in their attack range. A single ground or air anti-air unit can only intercept once per turn. Interception does not guarantee destruction, and it's possible that your unit will still be injured or destroyed even if the intercept is successful.

My strategy when assaulting a naval stronghold such as this is to bring overwhelming force, and plan to lose a few units. (This strategy would certainly be historically accurate...)

If air units are an issue, are you bringing in adequate ground-to-air (and/or sea-to-air) defenses? Carriers can store fighters that can intercept enemy fighters attacking your land and naval units.

I would suggest setting up some carriers and other ships to protect your embarked land units from enemy air and sea units while they get into position. It's not 100% clear from your question, but you are aware that you can attack the city while embarked, yes? Otherwise you've got another turn delay where your unit is vulnerable when you disembark.

As I suggested in the comments, I'd also consider choosing a different tactic or priority target if you can. Small 1 or 2 hex island cities aren't generally high value targets, and their production pales in comparison to a city near forests and hills. Plus, they're hard to assault (as you've noticed) and if the map isn't all islands, fighting the brunt of their forces in a sea/air engagement by building a bunch of ships you don't have a use for once you capture the islands might not be the best use of your military resources.

If you can bypass these islands, you might be better off capturing some of their higher value cities inland, thereby crippling their war machine. Who knows, they may even offer you the islands when they beg for peace... :)