None have a PC-architecture (x86+other HW standards), so no, they aren't. It would be possible to make a microcontroller that can, but nobody does. Intel did make a few experiments with x86 microcontrollers, but even those were afaik not PC-like enough, because you generally don't want a lot of those things in a microcontroller.
Generally a microcontroller has no MMU. A microprocessor may mean the same thing. The industry term for something with an MMU is an application processor (Cortex-A). There are other terms that can be used, but it will need to be specified.
There is some fuzzy area between MMU-less, small processors, and processors with no MMU but large enough to run Linux in a MMU-less way.
It's not very hard at all. The issue is exclusively licensing fees.
Cheapest depends. There are Allwinner parts that are $2? But with the move to RISC-V, we will see costs decrease further, probably so that a Linux-capable part is just as expensive or cheaper than an MCU.
Whenever possible I reach for a part that can run Linux, because most of any embedded task nowadays is connectivity. A lot of parts include a microcontroller on the same die for realtime tasks.
There are some recent x86 SoC's which basically implement i586 and have been made to run both MS-DOG and Windows 9x. Commonly used by retro- and vintage computing hobbyists. None that would cost $1, though.
Things called "microcontrollers" usually have these properties:
* They are actually CPUs, but with very weak feature sets related to desktop & scientific computing
* They may or may not have hardware to do any kind of advanced computation, starting with plain old multiplication and division of integers (addition and subtraction of integers is enough for what they do). Yes you read this correctly. The vast majority of 8-bit microcontrollers cannot even do division or even multiplication in hardware.
* They run with clocks from kilohertz to low megahertz
* They might be 8-bit, 16-bit or 32-bit, with rarity going up as the bitness goes up. There's a ton of 8-bit microntrollers, and 32-bit are much rarer. 64-bit ones are very very rare, since if you need that processing width, you might as well go for a single-board-computer like a Raspberry Pi.
* They are used to DIRECTLY drive other hardware, not in an abstract way. They literally regulate voltages on their pins to make stuff happen with connected hardware.
* They might have RAM space ranging from a couple of BYTES (yes, BYTES, see e.g. this one having 32 bytes of RAM: https://www.microchip.com/en-us/product/ATTINY9#) to a couple of hundreds of kilobytes. It's very rare to have a microcontroller with a megabyte or more of RAM, because, as before, in that case you might as well just use a full-featured desktop-ish CPU.
* They are usually lacking a special bit of hardware called a MMU (Memory Management Unit) which is used by "real" operating systems like Linux and Windows to manage process isolation and many other useful things related to memory. If you need a MMU, again, just use a SBC.
So where do we use microcontrollers? Well, what do you think happens in your garage or car keyfob when you click on the buttons? What tells your microwave when to finish and with which power level to nuke your food? What drives your electric toothbrush? Your car's gas valves? Microcontrollers, that's what.
They are currently (or have been a few years ago) so cheap we can readily add a microcontroller to every toilet paper leaf, and it would only bring the price of a roll up by a dollar or so. What's stopping us to do just that is the sorry state of current batteries and energy harvesting in general - those technologies simply are not as developed (yet).
So, no, you wouldn't describe a chip which can run Windows 98 as a "microcontroller".
