East coast for the first time

What you really need is a travel agent to plan it all for you. That's what they get paid to do.

See coalas and kangoroos (My dream since I was a kid)

Seeing Australian animals in the wild is a bit of a random event. Kangaroos are very flighty so they disappear very quickly if you come across them. Koalas are very sedentary they will be grey blobs of fur high up in the trees. I’d suggest going to somewhere like the Healesville Sanctuary in the Yarra Valley near Melbourne. Yes it is a zoo (but in lovely natural setting) but you can see up close a lot of native Australian animals such as kangaroos, koala’s and platypus. For an extra fee you can even pat some of them.

http://www.zoo.org.au/healesville

For the Great Ocean Road there are a lot of 1, 2 and 3 day bus tours out of Melbourne – do a google. Alternatively you could hire a car and do it yourself comfortably in 2 days.

With a car out of Melbourne you could also do a day trip to the Yarra Valley and include Healesville Sanctuary. Another trip is to Phillip Island for the penguin parade in the evening. There will also be day bus tours to do these places.

Maybe do some hiking in the nature where I'll be rewarded with a great view or a hidden waterfall/ lake

There are heaps of lovely short walks in the areas I’ve mentioned above.

http://www.visitvictoria.com/Regions/Yarra-Valley-and-Dandenong-Ranges/Things-to-do/Outdoor-activities/Walking-and-hiking

http://www.visitvictoria.com/Regions/Great-Ocean-Road/Things-to-do/Outdoor-activities/Walking-and-hiking/Great-Ocean-Walk/Short-walks.aspx?page=1&ps=10

That just about covers your week in and around Melbourne!!

An expensive, but maybe once in a lifetime event, would be maybe go up to Uluru.
Or if thats too far, maybe the Great Barrier Reef.
WA and Rottnest?
SA and the opal mines? Port Lincoln and shark cage diving?
NT and the crocodile jumping event?
Keeping to NSW, Blue Mountains outside Sydney? Broken Hill on the train to Adelaide?

O, if 1986 is your year of birth, and you are indeed from Germany, I would if I were you, if you could...
Consider the Aust WHV.
1986 if year of birth, means you are 29?
And 31 is the cut off.
Consider that one, thought if its too much of a side step consideration.

If you're driving along the Great Ocean Road, stop over in Lorne on the way there - there's some great hikes to beautiful waterfalls in the area: http://www.visitgreatoceanroad.org.au/the-otways-coast/lorne/adventure/lorne-walks

From the end of the Great Ocean Road, head inland to Halls Gap in the Grampian mountains - again, lots of lovely hikes, great scenery and views, a couple of waterfalls as well as loads of kangaroos that come out of the bush at twilight to graze as well as the chance to spot koalas (although they're tricky to see)... I saw an echidna when I was last there too.

Thank you for the tips! :)
I'm most certainly come back to Oz another time with more time on my hands and will do Uluru another time. Read on another forum that flights are quite expensive and I think it's too stressful to fly up there "just" to look at a big rock ;) when I'll go there, then I'll want to have more time and also visit some surrounding cities. So next tim! :)
Will google the other places but unfortunately the shark diving or any diving is a no go for me. Got problems with my ears and are therefore not allowed to go scuba diving. This is also why the Great Barrier Reef isn't on my to do list :(
Blue Mountains are already on my list ;)
Broken Hill sounds like a place from a horror movie! Hahaha kinda like "the hills have eyes". Will google it and I'm sure my imagination is just running too wild ! Hahaha
About the question if I'm German and born in 1986, yes I am. Since I'll only be in Oz for 25 days, I don't really want to work during that time. When I'm done studying this autumn, I might consider doing some work and travel in Oz. :) I also think that it would be too short notice to apply for the visa since I'll be flying there in less than 2 months.
Thanks again for the cool tips, will google all places and then decide :)

Wow, so many tips! Thank you so much! Will definitely check them out and then decide where to go. Or maybe all places :D

That sounds amazing! Thank you so much for the tips! :))

I saw an echidna when I was last there too.

I had to google that animal since I never heard of it! Haha! So if I spot one , I'll at least now know what it's called :D