The preposition iz would be more correctly translated as "out (of)" and s/z as "off (of)". Slovene, unlike English, uses two different prepositions to denote "from [place]", which forms a pair with two prepositions for "to [place]": iz - v (out of - into) and s/z - na (off of - onto). There are some rules so you can often predict which preposition a certain place name takes, but when it comes to many native place names in Slovenia, it's sometimes hard to tell and the best way then is to see what the local population uses.
Grem v Ljubljano. - Sem/prihajam iz Ljubljane.
Grem na Jesenice. - Sem/prihajam z Jesenic.
Grem v mesto/vas/gozd. - Prihajam iz mesta/vasi/gozda.
Grem na grad/goro/otok. - Prihajam z gradu/gore/otoka.
Grem v Italijo. - Sem iz Italije.
Grem na Finsko. - Sem s Finske. (quick rule of thumb: if the country ends in -ska, it requires na-s/z)
Similarly, the preposition od (from) forms a pair with do (to/towards). This is used in time tables, for example, od šestih do petih (from six to five), when expressing directions, or when saying that you received something from someone, or going to someone('s place), such as Grem do Mateja (I'm going to Matej's) - Prihajam od Mateja. (I'm coming from Matej's).
So, with that I hope you can figure out yourself which preposition should be used in the example sentences you wrote before, but if not (and if there's anything unclear, ask), here are the correct versions:
Sem vozil od Jurovskega Dola do Mežice.
To lepo uro sem dobil od Matije.
Se je smejal tako močno, da je padel s stola.