All the cores work with HDMI.
Yes you need an OTG cable plus a hub or an OTG hub because there is only one micro USB port for devices (there is also a mini USB for serial console). The DE10 Nano comes with a small OTG cable so you can just use any hub you have around. I don't use a powered hub but some devices might need power.
Not sure about a PS3 pad but rumble is not supported by any core. I will try when I get a bit of chance and report back. My Atari2600 adapter has some issues and I'm investigating them. Eventually it will work
I have plenty of controllers to try and I plan to update the firmware code if needed (once I become familiar enough with it)
There is only one type of memory board. Look at the dedicated pinned thread if you need one, some people here sell them.
The MiST style buttons (i.e. same functions) are on the IO expansion board.
It uses MicroSD cards. I use 32GB but it can probably use more. In any case its plenty of room.
For.a case I used the 3D print design from Negsol. There's a website that finds nearby 3D printers and you pay them to manufacture it for you.
Here's mine in action:https://www.instagram.com/p/BZYnq_YFjhL/
Here's the case Thingyverse files:https://www.thingiverse.com/thing:2527243