1) The PSU is really really marginal. Be SUPER careful how you connect things to it - 2 rail is a disadvantage for these things compared to single rail. Looking at the PSU specs, you should hope that the PCIE power connector is connected to 12V2. Make sure ALL other 12V stuff goes on 12V1. (or vice versa, depending on where the mobo/sata connectors will be pulling their power). The R9-270 TDP alone is a bit under 150 watts, 12V2 max is 18A x 12v = 216 watts (which actually makes no sense, as the PSU spec says that's the total of 12V1+12V2 - something is really fishy). Look really carefully at where the sata connectors is drawing power from - that's your other 12V draw although it's relatively minor (prob around 30w max), on top of the amount that might go through the motherboard to the video card. If you divide it right you can probably get by.
The 5v isn't so great either - 4570 has TDP of 85 watts, the motherboard is usually at least 70 watts, the total is really unliikely to be under 160 watts + you have to allow some room for the disk drives, and a bit less for the fans. Errrrr - if there's no surprises, you can *probably* squeak in, but don't even think of overclocking anything, ever. Even plugging in 1 more pcie card would scare me. Turn off everything in the bios you don't need - all those floppy ports, unused USB and sata, onboard graphics once you get your video card, etc.
4) The old card is an AGP slot card, it's not gonna fit in your new motherboard at all.
You might be able to get by with the onboard graphics for a while, they have been getting better and better. Nowhere near a decent card like the 270 though.
That's kind of an expensive board, you're paying for features you're not gonna use - onboard graphics/DVI/HDMI since you'll be adding a real card, SLI capability and 2 more full-sized pcie slots for starters. You could almost certainly save a few bucks with a less-featured version of that - a super quick look finds
http://www.amazon.com/Gigabyte-Intel-6Gbps-Motherboards-GA-Z87-HD3/dp/B00D94X6AK which is $30 cheaper.... Look at reviews, compare the features that you're actually gonna use.
EDIT: Also, do you live near a city with a Microcenter? They have pretty good cpu/mobo combos - for example you could get the 4670K + Gigabyte GA-Z87-HD3 LGA 1150 mobo for a total of $285 (plus tax, unfortunately)