I think the best answer to your question is the Intel Compute Stick.
To answer your feature requests:
It is small-scale being in principle a HMDI stick (~100x27x13mm)
It has a processor that is good enough (Atom Z3735F)
It has 8GB (Ubuntu version) or 32GB (Windows version) of memory which is expandable by the MicroSDXC slot
It has 1GB (ubuntu) or 2GB (...
You might of heard of it, but the Intel Compute Stick is a very small PC running an Intel Atom processor (Atom Z3735F). The standard model has 32GB of flash storage and though it costs $150, it comes with Windows 10, which is nice even if you do plan on installing Linux. You can buy one here. I actually use this one.
There are two versions of the Intel ...
There are several developers working towards this right now! Unfortunately most of these projects are still in the "Crowdfunding" or "Pre-order" stages.
One promising board, the "UP-Board" currently on Pre-Order: http://www.up-board.org/
16GB EMMC memory
Intel processor (Compatible with Windows 8.1/10, Unknown if compatible with Linux)
I agree with Firepower and SEJPM that the Intel Compute Stick is the route to go for around $150 but if you are looking for a cheaper version of that, the MeeGoPad T02 is a great alternative for $79 (The Ubuntu Version) and around $109 (The Windows 8.1 with upgrade to Windows 10 Version)
Both sticks have a Intel Atom Z3735f processor, 2GB's of Ram and 32 GB'...
I'm looking for something similar to this myself; currently I'm using a CHIP from NextThingCo; you'd have to write the software yourself for what you want, but it runs standard linux.
CHIP can connect to two wifi networks (or provide one). There's no ethernet, however.
CHIP has a TRRS jack
Push button to config wireless ...
Your best bet is one of the Rasbperry PI 2 or 3 is $35
The PI 3 does actually have built-in wifi, but no WPS button or activity leds. They both have a RJ45 for hard wiring with activity leds.
You need a 5V adapter,SD card, USB wifi adapter and shipping and handling will bring the cost up a bit more.
Give up on the WPS and wifi activity leds and save $10-$...
Sounds a lot like the USB armoury which I found googling for a remarkably similar, older product whose name I still don't remember. The feature list suggests you could emulate a device for communications with the host/
NXP i.MX53 ARM® Cortex™-A8 800Mhz, 512MB DDR3 RAM
USB host powered (<500 mA) device with compact form factor (65 x 19 x 6 mm)
No, such a thing does not exist. However, you can do something quite similar, even though it's not quite a PC on a USB stick.
If I understand your question correctly, you want to use one computer (which may be very old and slow) for it's screen, keyboard, and mouse, and use another computer or device to do the actual processing.
As an example: I use an ...
I am assuming that you want to connect the Intel Compute Stick to something that you can view the output of the device, correct?
First off, you cannot connect it into the HDMI port of your MacBook Pro. That's output only. That's like trying to connect the HDMI out port on a DVD player to the HDMI port on a game console - it's not going to work.
What you ...
Depending on how much you want to spend I would say Raspberry Pi would be the way to go as you will be able to install a nice OS that could be tailored to what you are after, or if you would farther have a slightly easier system there are a number of Zotac mini PC’s that pack a lot of hardware in a tiny form factor.
As for the controller if you have a look ...
You can buy used. For example, This - HP DC7800 Ultra Small Form Factor Desktop Computer (Refurbished) - is an extremely cheap x86 platform that does actually have some upgrade potential. It costs $50, but is more powerful than some SBC options and certainly more configurable.
It is the cheapest such machine I am aware of from large suppliers online. You ...
The Pi Zero makes this easy and cheap. Set up the Pi and follow this tutorial: https://learn.adafruit.com/turning-your-raspberry-pi-zero-into-a-usb-gadget/ethernet-gadget. I found that ssh is a good way to connect to the pi to perform these instructions.
When you are done, plug a micro usb cable into the USB data port on the pi and plug the other end into ...