Warning, OP might have been lucky with the weather, do not assume that one successful hike means that you can do it as well.
I never answered my own question here. Having done the thru-hike accross Iceland, I guess that now I'm qualified to make a response. The answer from @gerrit was imho overly pessimistic, so here is its counterpart.
I hiked the red itinerary here:
Here is the GPX track. I made a few blog posts about it on my personnal website (french).
I hiked the 420km in 13 days, with the first 3 days much shorter (~25km/day max) and the 3 last around 25km/day (I was on the Laugavegur, so I was sleeping near the huts). In the middle, I went up to 50km/day, because hiking on the desert is quite easy as the sand was often compact. It was mostly off-road tracks, but I often went off-track for up to 2 days.
For someone used to hike in the Alps (or other mountains), hiking in Iceland is mostly easier because of the lesser elevation. My thru-hike had a total elevation of 7500m, corresponding to a 4 days hike in the Alps for me. The difficulties were very localised, but could be very tiresome (I remember crossing 2km of lava field in almost 2h).
The main hazards were:
weather: I had 2 days with winds up to 130km/h. I had a few hours with a fog so dense I couldn't see 10m ahead. Without GPS I would still be there.
river crossing: definitely the most dangerous. As I was alone, I had to really be sure before crossing. If a big river was ahead, I had to make sure to cross it the earlier in the morning possible, because the rivers are stronger the afternoon.
solitude: yeah, being all alone in the desert isn't so easy.
ground: some lava fields were very tiresome, difficult and dangerous (easy to hurt a leg). Other time, the sand wasn't compact so it was also tiresome.
The safety gear I took:
a satellite beacon to call for emergencies: I didn't use it (fortunaly). I could/should have use the Gen Spot 3 (can send beacon by e-mail to family and friends), but the mobile network wasn't so bad so it was ok.
a phone : as I said, I could send a message or make a call almost everyday, so it was sufficient.
a GPS : so handy. When the weather is good, you can probably (and I'm not even sure) do without, but when the clouds kick in, it would be foolish to be there without a GPS. I had a Garmin eTrex 20.
trekking poles: absolutely needed for river crossing.
a good (homemade) tent: there is no shelter from the wind in the desert...
That being said, if your are prepared enough (good physical condition, knowing how to use your gear), I think that hiking in Iceland is not so difficult. But it's not for everyone either :)