So, apologies in advance but this will be a negative answer.
Normally I'd flag this sort of question as far too broad, but realistically your premise is for a world that's either going to turn back to a terrestrial state (and so you can just look at the history of life on earth) or will remain so hideously inhospitable that surface life likely won't exist.
Lightning sufficient to glass the planet's crust down to a depth of several km sounds like enough energy to sterliise the entire world, and probably boil off all the oceans, too. Seriously, that's overkill power like you'd get from a deathstar, not bad weather.
That aside, given enough time for life to arise, plate tectonics, volcanism, meteor impacts and general weathering would have done a good job of converting the glass back into sand and turning over continents to produce new swathes of igneous and sedimentary rock, uplifting coral reefs from the sea floor and so on. The places most likely to support life would be recently arisen volcanic islands, or recently formed lava flows. Such landscapes can be extremely fertile, and vastly more accomodating than raw glass. They're also quite terrestrial in nature.
"There's no tectonics or volcanism!" you might say. Well, that'll cause you a whole raft of problems, such as widespread melting of the crust when the world's inner heat can't escape. If there is no inner heat to speak of, it suggests your geomagnetic dynamo won't be providing much of a magnetic field anymore, and everyone will get zapped by cosmic radiation and then your atmosphere will be depleted by the solar wind. It also removes the carbonate-silicate cycle, which will make it rather difficult to get the CO2 levels of your atmosphere down, meaning that any life that does arise will be very different to that which arose on earth.
What would flora and fauna be like in a world that was made of glass?
Probably quite primitive, but over time the world would get less and less glassy and more and more terrestrial, at which point you might get something that looks like terrestrial life, I suppose.
Lichens might be the most plausible initial colonists, descendants of algae and photosynthetic bacteria and the like that can survive drying out in the intertidal zone. Unless the world was substantially more cracked, broken and rocky or sandy, more complex vegetation seems unlikely to get a good toehold, as bad weather would blast it clean off a hard and unforgiving substrate.
If/when the land gets more eroded by weather and biological action, the glass might break up a bit and allow for more plant-like things to anchor themselves in cracks. Damaged land seems like a pre-requisite for larger plant life, and larger plant life will damage the land further and eventually form soil and so on. The more the environment becomes capable of supporting complex life, the less like glassworld it will become.
It occurs to me that your oceans might not be saline, depending on when the glassing occurred in the planet's history. The nutrient balance is going to be quite different too.
Would they carve shelters into the glass, or not use shelters?
Sounds like so much work for primitive species that might not even have tools that they just wouldn't bother. More hospitable terrain would arise naturally though.
One of the ideas I had was that they would primarily get around with glass skates (ice skates, but on glass), but would that even be possible with standard materials?
The surface of ice is quite a complex thing. At sensible temperatures, you'll find it readily melts to provide a very slick and wet surface. With suitably hard blades and spikes, you can dig in to the upper layers of ice, enabling skates to work without just skidding out sideways. Glass is much, much harder than ice, so you'll never get any useful purchase with skates. You'll also find that weathering over time will damage the surface of glass, making it less and less smooth, and it'll never reach the temperatures required to re-melt it to get a smooth surface back again.
In short, what would the flora and fauna of a world where all land was entirely glass be like
Sterile, I'd guess!
and how would humans live and behave in this glass world?