Upgraded iPhone, Library and Syncing problems

Hi all, so here's a run down of the problem.

I have a MacPro with a very large iTunes library, a fraction of which fits on to my 32GB iPhone 4. The vast majority of this content is ripped from my large CD collection, only a little is purchased from the iTunes store. A little while ago my MacPro developed a malfunction which I believe is a graphics card failure so it now won't boot. I'm studying full time again and cannot afford to get this fixed yet so since it stopped working I've not synced my iPhone 4 (and can't).

My iPhone 4 has been on it's last legs for some time now and I was recently bought an iPhone 5s to replace it. I've got the new nano SIM. I have a 50GB iCloud account which I hoped 'd be able to back up my entire iPhone 4 to in order to transfer everything to my new 5s. Obviously I've recently discovered iCloud will only back up iTunes purchased music which, while being annoying, I can live without my entire library for a while until I fix my MacPro, but I really need to start using my iPhone 5s as the old phone is basically unusable due to slowness and terrible battery life.

So I backed up everything I could from my iPhone 4 to iCloud, popped the new nano SIM card in to my new iPhone 5s and tried to install the back up on to it from iCloud. It seems to work but then gets to a screen where I key in my passcode, the screen then goes totally white with "Swipe to Upgrade" at the bottom, but it won't swipe, the phone basically becomes unresponsive from this point onwards. Very frustrating.

My girlfriend has an iMac on which she's never used iTunes so I'm now thinking I could use this to get my new 5s up and running but I still need to sync my iPhone 4 to it first, is this possible? I know I'll probably have to sacrifice my music/videos until I fix my MacPro but can I at least get everything else from my phone on to the iTunes software on my girlfriend's iMac and then sync it to my new 5s?

One other very strange thing: while trying to transfer my stuff from iCloud on to the new 5s I used my own iCloud account, obviously, but at one point it asked me to enter the password for an Apple ID which was my friend's email address, not mine. I have never synced my iPhone to his iTunes or used his account details to ever purchase anything from iTunes so how on earth has this happened? I'm utterly baffled. It gave me the option to "skip this step" which I did because I don't want my new 5s syncing to his Apple account and besides, I have new clue what his password is even if I did want to.

I've been an Apple user for some 11-12 years now...since before their products became 'must have' fashion items for everyone regardless of whether they actually need the kind of software or power that Macs used to be purchased for (music/video/media/photography etc.), I realise this may make me something of a dinosaur but they used to just work, and work very well and very reliably so this kind of situation and too-ing and fro-ing and incompatibility and constant need to update is absurdly annoying...just sayin!

If any 'new' users who know, better than I do, the ins and outs of how I can make my new iPhone just work, I'd really appreciate your advice.

Many thanks