Pat,
You could have a few problems here but, I don't think the starter is to blame. Usually, the starter will only stay engaged too long because it is still receiving the signal to run from the ignition. You might have a bad electrical ignition switch. Notice, I said ELECTRICAL switch. If the key doesn't feel right in the cylinder, you might have a bad cylinder too. You also, might have a bad neutral safety switch. This would cause an intermittent no crank condition. You might have a battery on it's way out.

I don't have an answer for the rough running after start-up but, it won't have anything to do with the starter or related starting stuff. You might have an intake leak.

As far as the humming noise, you might have gremlins under the throttle cover. There should'nt be anything running under the hood if the key was out. Are you sure you didn't have a headlight wiper motor running?

Pete

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If you can't afford to take your car to the dealer for maintenance, then sell it. Don't cry when the local shops can't deal with the technology anymore.