I've also been experiencing sync issue between my nexus 4 and live.co.uk account, set up as exchange on my device.
Been in contact with M$ and got about as far as you guys have (nowhere), so have been doing some hardcore testing over the past couple of days to try and find out from where the problem stems.
Like a few of you I have excluded using the android 'outlook.com' app from +SEVEN as it is both ugly, not particularly functional, and prevents the editing of contact details from the native People app on the device.
So after a monsterous amount of syncing, adding, removing, syncing, editing, syncing various contacts between three places (outlook.com, my android handset, and exchange 2013) under various circumstances I THINK I've found a 'solution' that works for me.
If after syncing the account all I do on my handset is add a contact, it works perfectly and syncs.
If I amend 'conventional' details (name, address, phone) it works fine.
If I delete a contact it works fine.
HOWEVER
If I add a picture to the contact, or click the star to add the contact to my favourites no contact modifications then sync either TO or FROM the handset.
This seems to be holding true at the moment, but I will continue to monitor the situation and update this thread if it seems not to be the case.
If it does hold true I can only assume that during the sync process android tries to deliver fields that exchange doesn't support at which point it freaks out and refuses future requests. This would potentially explain why the outlook.com app doesn't allow
for updates from the handset?
Anyway - I hope this helps some people.
If it does transpire this is the issue it'd be nice if Microsoft could maybe re-assess their EAS protocol to perhaps just ignore any sync data it doesn't like rather that causing an outright failure.
Future update to follow.