Yes, your PSU will provide the power connections, and you do need one SATA data ribbon (7-conductor is standard) for it to connect to a mobo SATA port. If you are buying components new, it is normal that the mobo will come with several accessories, including cables such as this to connect your peripherals. So check the details of your mobo to be sure it comes with a couple of them - say, one each for your HDD and optical (SATA) drive. Check details of your case - that usually comes with a selection of mounting screws you need to fasten the HDD into the mounting cage.
Unfortunately, it seems there is a non-answer. The newegg page does not talk about anything so small as accessory cables. Their link to the ASRock product page shows under Specifications that the 4 SATA data cables come with it OPTIONALLY, so I guess that it is NOT guaranteed to have them - probable, but for sure?? If you want to be very sure and risk ending up with too many of the, order a couple of SATA data cables - the ones with locking connectors on the end are recommended so they don't work loose. The cost $2-$4 each.
Yes, those will be fine. They are called "SATA3" cables, more properly SATA 6.0 Gb/s data cables, but the truth is that all SATA cables are the same as long as they are of decent quality. Any claiming to meet the fastest requirements ought to work in all SATA applications. You'll need 2 - one for each unit - but sometimes I order an extra so I have a spare on hand later.