I know very little about this game and have no intention of playing it, but I am very curious how chat will work on console.

I played WOW for along time and it had many different chats such as friend, guild, trade, local defense, whisper, general, battleground, party, raid and normal.

Even if ESO has one chat I am still curious about how it will work since everyone uses mics.

I am also curious if you will be able link quests, items, achievements into it?

But to answer your questions:

All consoles since, like, way back, have had keyboard compatability. I can't imagine the PS4 and XB1 won't have it also. So basically the exact same way as PC.

And for the second question, I'm sure you can yes.

Also they might design some sort of interface for those without a USB keyboard for their console to either have a virtual keyboard appear on the screen, or some sort of menu driven chat that allows you to choose from basic things like "I am looking for a group" or "I want to sell an item", etc, though that would be so limiting I can't really see that being the case.

Current and future consoles have usb ports for recharging controllers and many other functions, like plugging in a usb keyboard. It's not hard, or expensive to find a cheap usb keyboard to use with mmorpgs like ESO.

I've tried using the virtual keyboards provided with console mmorpgs (Final Fantasy 11, Everquest: Online Adventures) and it's really annoying trying to have a conversation that way, or using phrases via a hotkey system. Just do yourself a favor and buy a usb keyboard.

Just curious, if you are not interested in playing, then why are you even asking this question or even care?

Anyone, yeah USB Keyboards.