Specifying the failover partner when using Availability Groups
There's two parts to this question:

First, can you use the Failover Partner connection string tip with AlwaysOn Availability Groups?  No.  AlwaysOn AG's "Listener" technology is the replacement.  Have your connection strings point to the listener name and they'll always get the primary replica.  (For the next part of this answer, I'm assuming you're using the listener name - if you're not, start, heh.)

Second, why do some queries fail to connect to the listener?  This has to do with the number of DNS entries for the listener.  All possible subnets for the listener will be in DNS at all times.  If you've got a listener in 192.168.1.X and another in 192.168.100.x, both listeners will always be in DNS.  By default, your clients will try connecting to each of the DNS entries serially, and not always in numeric order.  If you've got a 30-second connection timeout, it's possible that your app will only try one of the IPs and then fail before it has the time to try the second one.

If you want to try connecting to all possible IPs simultaneously, check out the MultiSubnetFailover = True options for the SQL Server client as described here: http://msdn.microsoft.com/en-us/library/hh205662.aspx

Otherwise, you'll need to increase your connection timeout to account for the multiple IPs.

Update Feb 27: the question added, "I was therefore trying to come up with a way to avoid manual intervention for these (replicated) DBs should a failure of the replica occur and the other databases which are part of an AG failover."

Ooo, unfortunately, no, if the databases aren't part of the Availability Group, you're going to be doing manual work in order to fail over. One popular option is to use a DNS record, and just repoint the DNS record at whatever server is currently hosting the primary copy of the databases.