Every time i use my "force push", it sounds like my char is puking instead of uttering the dragon-words
which enables the power of the pushforce, why is that?.
A friend of mine said something about him using 3 different buttons when he uses the shout, one
for each word of the phrase. But i only see 1 button, 'z' , for activating shouts.

Also. Is the Push-force phrase the only phrase which has 3 words in it?. Because i have scoured
almost the entire map and haven't found any extra words for increasing the power of Firebreath
for example.

IIRC Firebreath is taught to you. And to charge shouts, you just hold down 'z', not tap it like you would if you only wanted to use 'fus'. All shouts have three words in them, some shouts you learn them all at once though.

Yep, just hold the button down longer to get all three words out. All the words are out there somewhere you just have to find then and it's a BIG world. If you want to be spoiled as to their locations just follow this link:
http://www.uesp.net/wiki/Skyrim:Dragon_Shouts

Fable much?

Like others said, if you hold down the button/trigger the Shout will activate on it's own when it's fully charged. I haven't started a Mage, but as I understand spells they are activated when you release - the shout will go off automatically when it's fully charged so make sure it's aimed before you start it. The Unrelenting Force ("force push" ) seems to have a fairly narrow 'path' (even narrower than Fable II/III) so try to align multiple targets in front of you as best as possible. When you have all 3 words for it it will interrupt and stun.