One easily can find slides with assembler for x086 processor. But does modern Intel CPUs works the same way?
Yes, basically. Modern CPUs have a bunch of added instructions and modes and memory management and stuff, but you can write program pretty much the same way you did back on the original IBM PC. In fact, I've done it recently. Most PC boards even emulate most of the ancient hardware.
Can I to plug a cpu in a breadbord or have to use a special socket? Other hardware is needed?
Most desktop-class CPU chips requires a set of three chips. The CPU itself, a "Northbridge" chip that is essentially a memory bus controller, and a "Southbridge" chip that does most of the standard IO. These days each of these has a couple hundred pins, so it's a bit past most "breadboard" capabilities.
I don't know that anyone has recently tried to build a minimalist x86 system with a full-scale CPU. No one seems very interested in running a 2GHz CPU at a couple MHz with tiny memory, given that you can buy an OTS product that does a lot more for $100...
It does look like there are still older smaller versions of x86 CPUs being made and sold. Digikey has an Intersil 80c88 and a bunch of 80186-class chips listed.
There is some speculation that with the original x86 instruction set patents expiring soon, there might be a flood of x86-compatible chips showing up from ... everywhere (possibly including microcontroller versions.) But I haven't seen this actually happening yet...
x86 based CPU's are so complicated that programming them the way we do with Arduino is not going to work.
You need loads of RAM and storage memory to be able to do things the way you want to.
I disagree. A "small" x86 microcontroller with (say) 128K of flash and 16K of RAM (similar to some mid-range ARMs) would be a very interesting product, capable of competing directly with AVR/Arduino if priced appropriately, and programmable using pretty much similar tools (gcc, etc.) So far, Intel only seems interested in attacking the highest-end ARM chips (with the "Atom" line.)