I had my prayer corner (not in the corner as we have no east facing corners in my house.) - or the beginnings of it set up on a dresser on an east facing wall.
Fwiw, my icon corner faces Southeast rather than East.
...is there a specific order or placement of icons that should be followed?
Other than Jesus being on the right and the Theotokos on the left (if they're paired together), I don't think so.
As I said above, my prayer 'corner' is a flat wall - the nearest corner that I can use is South East. . .better to stay on the East wall or should this really be in a corner?
I'd say it's really up to you--whatever makes it best for you to pray. If it'd bug you one way or the other, I'd say go with what works best for you.
Are there any necessary icons that I should think about purchasing that should be in any any icon corner (like on the Iconostatis at church?)
I don't think so, you've got (or are getting) the ones that I'd have said were at the top of the list. After that, it's up to you, though it's nice to have a reason for the icons you choose after that. For example I have St. James to represent the Biblical writers, St. Gregory the Theologian to represent the Church Fathers, etc.
I have an incense burner, and vigil lamps. . .I will order a hanging vigil lamp as soon as I am able. Is there anything else that I am missing?
Perhaps keep some candles handy, in case your priest ever wants to use them when he visits. That's all I can think of though.