Backwards Compatibility is just a term that describes emulation of predecessors.

Emulation can range from running entire virtual operating systems to excuting simple predecessor instructions.

There are two types of emulation, software and hardware. Hardware emulation requires the actual components to be in the system, this means you can A) build the newer model using similar technology, which holds the system back if the architecture is too outdated or B) include the old architecture along with the new one, more expensive. Software emulation requires architecture that is alot stronger then hardware its emulating because it virtualizes the hardware in its own, and that virtual hardware runs the software. Typically, emulators run the whole OS, but well optimized emulators can run barebones and stripped OS. E.G, my phone can emulate some PSP games at native resolution and framerate despite only being maybe 5 times stronger.

Long story short, hardware emulation is the only viable online solution for emulation and subsequently BC, streaming can do the hardware emulation by running it on a more powerful server, but their is the issue of network dependency.

A lot of people mention ARM here. Can't the next console use ARM hardware to emulate the Wii U's hardware since the Wii U does use some ARM components? Then it wouldn't have to hold them back, right?

The issue is that it use a power pc CPU, the ARM components aren't essential for gaming and the GPU's more or less easy to substitute with another, but you can't emulate Power PC with ARM of the same spec, its a different instruction set.

Backwards Compatibility is just a term that describes emulation of predecessors.

Emulation can range from running entire virtual operating systems to excuting simple predecessor instructions.

There are two types of emulation, software and hardware. Hardware emulation requires the actual components to be in the system, this means you can A) build the newer model using similar technology, which holds the system back if the architecture is too outdated or B) include the old architecture along with the new one, more expensive. Software emulation requires architecture that is alot stronger then hardware its emulating because it virtualizes the hardware in its own, and that virtual hardware runs the software. Typically, emulators run the whole OS, but well optimized emulators can run barebones and stripped OS. E.G, my phone can emulate some PSP games at native resolution and framerate despite only being maybe 5 times stronger.

Long story short, hardware emulation is the only viable online solution for emulation and subsequently BC, streaming can do the hardware emulation by running it on a more powerful server, but their is the issue of network dependency.

They can't put the old Wii U CPU in there strictly for BC and build the ARM console from there?

That was the issue with the Original PS3. It really depends on how cheap they can make the Wii U before the next console comes out, and it will run its CPU for Wii U games but it might not be price effective.

So would it be possible for Nintendo to make an ARM console as powerful as the PS4 for $300 with backwards compatibility to the Wii U and/3DS eshop by winter 2016?

They can't put the old Wii U CPU in there strictly for BC and build the ARM console from there?