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they should all go through. although it might not be automatic it may wait until you log into gmail before it looks at the cox emails. that is you are not actually telling cox to forward everything that comes in as soon as it gets it. instead you are telling gmail to pull it off the cox server. so if you look at gmail at 5 pm it will pull everything off. if you then look at cox directly at 6 pm you will see everything that came in between 5 and 6 pm.
that is because the outlook stripped it off of the server. go to tools and accounts and under advanced check "leave a copy on the server"
you need to decide where you want the permenant copy to be.
either outlook or gmail and set the other one to leave a copy on the server. the same setting is available by editing the pop mail settings for forwarded accounts.