Proper reclaimed sleepers of ye olde variety creosoted to their eyeballs can probably sit on the concrete and should behave themselves. Modern sleepers will still need treating. As TeePee says some sort of DPC barrier is usually recommended to stop direct wetting of timbers, especially softwoods.
I imagine the person who did your slab used the timber surround as permanent formwork and the guides to skim the surface level. The timber surround, being lower down and in contact with the ground, will probably suffer before any shed/lean to you build on top of the slab tbh.
The youtube video makes a lot of use of a nail gun, which is fine, provided you never want/need to take it down and change bits. Personally I go for brackets and screws etc. Takes longer to put up but I know I can have an easy time of it in future if I need to undo it. I'm currently trying to remove some noggins which were skew-nailed via nail-gun in the 1980s. It's not easy.
By using screws I also don't have to worry about inadvertently shooting a nail into myself/someone else. Getting the power setting wrong can have catastrophic effects. I recall one site having a major incident where one of the operatives was nail-gunning into a concrete wall, only one part of the wall was thinned down and when he reached that section the nail went through the concrete and into someone working behind it. Not pleasant.
Consider how you're going to fix the shed down into the concrete. Is the slab reinforced? What edge spacings and embedment depth do your proposed anchors require, and therefore what are the implications for where the shed walls need to be positioned?