In reply to Jimbocz:
What does your wife edit photos for? Just for fun to share online? For printing out at home? For professional reasons?
Depending on how serious she is, getting a monitor with the correct colour gamut can be very important. Here's a link with some info on the topic:
http://compreviews.about.com/od/monitors/a/LCDColorGamut.htm
If she gets a monitor with expanded colour gamut, and one that has a good colour temperature (some LCDs use shitty backlights that give a very high colour temperature, if I remember correctly it's better to have a low colour temperature backlight to better match natural light) - then it quite possibly will help her edit photos better.
Regarding 4k. It's not super necessary. The fundamental thing to consider is that the human eye has a mimimum discernable pixel size - the size of the smallest pixels you can see, before they are too small to see individually. In other words if you have a 50" 4k TV, the size of the pixels will be bigger than that of a 22" 4k display (obviously). But less obviously a 22" 1080p LCD might have a pixel size smaller than your 'minimum discernable pizel size', and thus the 'bottleneck' is eyesight rather than the monitor.
I'd definitely buy a graphics card - they're not that expensive, and make your computer a lot quicker by delegating 'manual labour'.
Best size depends on how big the desk is, how far away the user will be, and required pixel density.
I'm not up to date on best connections, so if there's anything better than HDMI I don't know. It's worth reading up about differences between the different versions of HDMI, as there might be things added in later versions that are worth having. From memory most of the changes between HDMI 1.1 and 1.3 were audio stuff so unimportant, but there might be more goodies added since 1.3.
Post edited at 14:30