Your question's title: "Is building giant ship-cities in a water world a good idea?" demands an answer that's sort of orthogonal to the actual question you elaborate, and to the answers you've gotten so far.
This rogue answer: Creatively, it's a brilliant idea. In terms of storytelling potential you have amazing scope if you can get this right.
In terms of contextual humans-vs-nature stress, this amps up the storytelling tension. The sea can always provide plot-driving difficulties and drama.
I've been there personally: I was a working sailor (submarine, tugboat, and bluewater sailboat.) Scary things can happen in a big hurry.
On the other hand, sometimes trouble can take a long time to build. When a big storm is coming, you have time to wait and worry and doublecheck and trice everything down... and then watch it start to happen. This is some of the best dramatic tension you can get.
When it comes to human-vs-human conflict (not just in the obvious sense of physical combat, but social, political, psychological, and familial competition and antagonism), your opportunities are huge. You have people doing the best they can in a tough situation. You have individuals of piratical mindset. You have possible conflict for scarce resources.
After all, protagonist/antagonist conflict is the heart of just about any story.
As a beautiful, harsh, and dramatic setting, the image of rafted vessels out at sea, establishing a survival society when the land has become unavailable, is really a great one. The sea is both beautiful and terrible, and ships in the kind of situation you describe have a haunting power all their own.
There. A powerful, beautiful creative concept. Just so we're clear. :-)
Now, to, you know, actually answer the question you're asking...
Previous answers have done a pretty good job of considering practicalities. I'll add my US$0.02 on some of these:
"And of course with no dry docks and no repair facilities it will be only a matter of time until your ship will sink, one after another."
This is an extremely important consideration. Until you've seen it at work, it's hard to grasp just how incredibly corrosive salt water is to steel. Since we're talking steel ships here, it's very hard to see how this concept of building your infrastructure out of tankers and aircraft carriers would be sustainable.
You could posit some kind of floating dry dock that would be a part of your flotilla/raft/floating city, but that's got a couple of severe drawbacks.
First, it's a terribly difficult engineering problem. All of these big ships are built and refitted in huge-ass shoreside graving docks. A floating dock big enough to accommodate these huge ship hulls is not very practical:
It has to be even bigger than the ships it services;
It will probably have major seakeeping issues. Large oceanic waves would create twisting stresses on the comparatively fragile hulls characteristic of floating drydocks. Making them big enough to accommodate huge ships will make them fail and break up in short order.
Second - and most fundamentally important - floating drydocks only kick the can down the road, because they themselves are subject to rust.
This is perhaps some virtuous Worldbiolding.SE bait. Hopefully some of the good people here will want to come up with some good answers. However, until this one is solved, the entire project is doable only in the short run.
Of course, if that's OK for your storytelling purposes, it becomes a feature not a bug. :-) MOAR DRAMA is good.
Well rafted ships survive better
Rafting does two important things. One is that it decreases the distance any boat can travel before smashing into another boat. This is counterintuitive, but it works like leaning into a punch in boxing. It decreases the distance the other boat has to accelerate before it hits you. Your boat will be peppered with smaller hits, but hopefully avoid a fellow boat being smashed into you.
Yes, absolutely. A good description.
However, it's not all fair skies and following seas.
Rafts are good for big waves, up to a point:
A raft of vessels has one huge advantage compared to single hulls when the waves start getting big: A raft is flexible. you can have a raft that covers acres and it will conform to the surfaces of even immense waves.
That said, there's a limit: If the steepness of the wave face exceeds the working tolerances of the vessel-to-vessel flexibility of the raft, there would be a catastrophic failure. I'm not conversant enough with naval engineering to make any suggestions about how that limit is computed; but it seems likely that gales and hurricanes would exceed the integrity of big ships rafted together, no matter how cleverly it was done.
And then, there's also the occasional Rogue wave to worry about... really, rafts of large vessels in the open ocean would not survive too well, over time.
Rafts are awful for big waves:
The other thing about rafts, which can be a huge disadvantage in a heavy seaway, is that they cannot maneuver as a unit. What this means is that the ability to turn your bow into the wind/waves is no longer there when you're rafted. Each vessel is going to have to take the waves however they come:
dead astern (not too bad),
off the bow or on the quarter (worst forces on the cabling joining adjacent ships), or
on the beam (very bad.)
Another concern: an independent ship under way has the option, not just of maneuvering to put the bow into the wind, but of turning to run before the storm. If you've got a 90 knot wind, and your vessel can make 10 knots, then running downwind you are only dealing with an 80 knot wind.
That might not sound like much, but it's actually a pretty big deal in terms of moderating the forces on your vessel.
(Note that this may not be good: sometimes running downwind in a storm will change the way the waves stress your hull in a bad way. Still, any ship captain will tell you that it's a good thing to have the option...)
All in all, I'm starting to have serious concerns about the feasibility of your idea.
I think that the fundamental creative vision is (as mentioned above) pretty wonderful. However, strictly interpreted, your particular scenario - lots of very large ships rafted together - seems unworkable.
There are some things you could do to fix it from the standpoint of practical shipbuilding and seamanship, if you're willing to relax some of your requirements. For example:
Raft dedicated components rather than big ships
It's not at all hard to devise modular floating structures that would raft together nicely. Because they wouldn't have the initial design constraints as independently operable ships, they could be incredibly superior for rafting purposes.
Really, these could be little more than floats with strong frameworks. Probably with decking or other minimal superstructure.
You could raft them together to create very rugged floating docks, even ones that sprawl out over a lot of area. It would be possible to dock smaller vessels at such a raft; and therefore possible to have the tankers and such keeping station (or remotely ingeniously moored), with small craft going to and fro. This would leave the big ships free to move away and operate independently in the event of a storm. This would preserve that particular conceptual element (of which I remain very fond.)
They could be built of a range of materials that wouldn't be as vulnerable to rust as steel hulls.
They could, and should, be small. They would accordingly not require the same massive drydocking capabilities as preexisting tankers/aircraft carriers/container ships.
This would make an excellent at-sea rendezvous/port for inexpensive small sailing vessels.
Consider semi-submersible large vessels/structures
Semi-submersible hull design is capable of amazing stability in high seas. This is achieved by having all significant buoyancy in the craft concentrated in underwater pontoons. The buoyancy provided by the pontoons remains the same when big waves pass overhead: submerged is submerged.
The pontoons are joined to a free-standing superstructure that remains above the water's surface at all times. Superstructure and pontoons are joined by strong struts that
do not show much net buoyancy change as the waves pass among them;
do not provide much of a surface for waves to break against, minimizing wave impact and the dynamic stress of wave action against the vessel.
I have personally seen this principle in action. Working on a submarine doing scientific experiments with FLIP, I was lucky enough to see FLIP holding dead still in heavy seas. ("Lucky" in relative terms. We, diesel-electric and therefore operating on the surface most of the time, were rolling all over the damn place.) There's a bit of footage in this video starting at around 3:40. Yes, it really was like that: FLIP holding stock still in big swells.
Made a believer out of me. :-)
Yes, you can do this, sort of.
You need to make at least the following changes to your premise:
Don't raft big ships together in open ocean. Have them together as a flotilla, working together, acting as motherships for smaller working vessels.
Treat the oil-tanker floating city/colony as a temporary thing, working towards something that can be sustained over time. That in itself could be a dramatic decades-long process.
A set of modular rafting floats, with some semi-submersible vessels at the center, could, over the long term, be the kind of floating city that you are asking about.