Just to start off, the "I" in ILEC is for "incumbent" and the "C" in CLEC is for "competitive". That is intended to imply that the ILEC is the one that's "always" been there and the CLECs are coming in to take over some of their business.

To give you some more history that might help understanding this: Long ago, there was a single telephone company in the US: AT&T. It provided absolutely every service that you could get. In 1984, the US government decided that they could no longer have that monopoly, so they split the company into a number of pieces. AT&T would keep the long distance services, but other companies would be created to be the local carriers, called RBOCs or, colloquially, Baby Bells (due to AT&T being known as the Bell System, after Alexander Graham Bell). Seven RBOCs were created from AT&T's local holdings, one each for the southeast (BellSouth), the midwest (Ameritech), the mid-Atlantic (Bell Atlantic), the northeast (NYNEX), the west coast and southwest (Pacific Telesis), the southern central (Southwestern Bell), and northern central (US West) areas of the US. Those companies essentially had monopolies on those areas until many years later. At the same time as all of this, a few new companies emerged to compete with AT&T in the long distance market.

Since then, more deregulation and, thus, more competition, has emerged. Now virtually any company can provide any service, but the land lines are still owned by those original seven companies (except that they've been bought out and merged and split again numerous times since 1984, so few of those original names are still in use).

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* ILECs are bigger companies than CLECs

Generally speaking, ILECs are the biggest companies in the market, but an ILEC in one region may be a CLEC in another.

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* ILECs are for connection many CLECs together (long distance calls)

No, they are effectively the same thing, they just happen to own the lines that both ILECs and CLECs use. "LEC" stands for "local exchange carrier". All LECs provide local service and that's it. Long distance carriers are other companies altogether.

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* DIDs are controlled by ILECs (why? thought the clecs would get e.g. 1-555-xxx-xxx and provide the rest themselfes like a net with a netmask for IP)

I'm not sure how DIDs are initially determined, but it's very uncommon for a phone in a region to not have the same area code as all the other phones in the same region. Although some regions do have more than one area code assigned.

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* A CLEC provides just one city (more or less)

More than just a city. Usually an area spanning several states.

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* A CLEC has it's own physical wires (last mile)

Not usually, if ever. CLECs "rent" those lines from the ILECs.

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* ILECs can rent long distance wires from other (bigger?) ILECs which have their own wires?

ILECs own all the local wiring in their region. They are (logically) connected to all of the long distance providers who work as interconnects between all the LECs. I don't know how it's physically set up.

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* A CLEC can be an ILEC, too.

A CLEC providing service in a region other than its own is a CLEC in that other region.

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* People pay $35/month for their telephone including city calls (everything which is supported by their CLEC and not using ILECs for long distance calls)

To be clear, all of the LECs provide nothing more than local service. Any long distance service is provided by a long distance carrier, which is a separate service. (To make things more complicated, the LECs often actually will provide long distance service, but usually at exorbitant rates.)

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* Cellphones can have free long distance calls (at which monthly charge?)

I've not heard of a cell phone plan in ages that didn't provide free long distance. There are a huge variety of plans. Most of them are a flat monthly fee for a set number of airtime minutes, plus some fee for each minute you go over. A typical one might be $60 for 900 minutes.

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* You can do a long distance call even when you call to the same region code and you can do a city call when using another region code (like New York city).

Yes. It basically depends on the phone density of the area. Calls within the same city are probably always local calls, even if the city is dense enough to have multiple area codes, whereas a sparsely populated region might only need one area code to hold all the numbers for a huge area, but calls to a place far away may be considered long distance.

This brings up the question of how it can be long distance even if it's the same ILEC providing service. I honestly don't know how that's the case, or quite how it works. All I can say is that it is.