The hx 620 is a single rail psu with 3x 18 amps @ 12V virtual rails (each 12V rail is over current protected @18A = 216W). (have a look at the sticker).
Each PCIe slot needs at least 75W @ 12V (6.25 amps). So my guess is, you are good with 3 RX480/470s. If your psu can handle >225 watts per rail than four cards, but I wouldnt recommend it for 24/7 usage.
Which card you are gonna use is absolutely your choice.
With powered risers, every slot should work fine, if it isnt shared with some onboard stuff -> have a look in the manual for that.
Edit:
Here is the test. They managed to squeeze 670 Watts (@12V) out of it. So with 4 RX 480 cards it is maxed out and sucks 900 Watts at the wall.
I agree with flikflak in that your PSU should be able to support 3 RX470's comfortably even at stock settings. With BIOS mods you could possibly get 4, but you would really be pushing it. My personal comfort zone is to only run a PSU at ~80% of its rating for 24/7 use, so using this guideline would net you about 500 usable watts.
At stock settings you would be using about 450-475 watts with 3 x RX470's. BIOS mods could drop this down to about 380 watts giving you just about enough headroom to add a 4th card, which would bring the total to around 520 watts or so. A bit higher than I would like to see, but considering you already own the PSU you could do some tests with three cards at first to see where you are at.
Get a Kill-a-watt or similar meter that can measure the peak wattage over say a 24 hour period "at the wall" to get a good idea of what you real wattage usage is. This will vary over the course of the day as heat and fans ramp up and down, so you just don't want to base all your math with a low reading you might get when your first start-up.