Yes it will run all games on ultra.
Yes it will fit in the Motherboard - the cards vary in size so check the length of the card by getting the manufacturer dimensions and check your case space. in most cases they fit just fine
4k gaming in my humble opinion is not worth the money you need to throw behind getting the horsepower to run games 40 - 60 fps. e.g 2 x 970's or 2 x 980's in SLI. Monitors are comparably a bad investment at this stage in terms of value.
If you can afford it, go for it, if you are average joe, a GTX 970 and 1080p monitor are plenty - you could go for a 1440p 144hz monitor but then again you may need more GPU horsepower maybe a 2nd 970 for ultra.
hope that helps.