[tardigrade]

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More recently I get burned a lot by people making straights on garbage hands

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It happens ... I wouldn't worry about that too much. Just make sure you are charging them the most you can when you are ahead and pay them as little as possible when they make their miracle draw.

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One area of play that I have never seemed to figure out is what to do when I have called pre-flop and then there is a raise behind me.

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I usually will call a pf raise behind me except under fairly extreme circumstances. If you had odds to play something like 44 or 98s before the raise, you have as good odds after the raise, essentially, minus some implied odds. On the other hand, if you played something like ATo, then you may be playing a dominated hand. I still generally call unless the pot is quite small, but proceed with caution after the flop.

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Basically, can someone be a profitable player without playing the pot odds down to the number?

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In-game pot odds are always an estimate anyway. You will almost never know your true odds, so the difference between 10% and 12% may not be critical. If you know most of the common situations like flush and straight draws, trips to full house, etc. and can get a quick rough count of your outs and use the 4/2 rule in a pinch, you should be close enough. What will ruin you is if you overestimate the value of your outs with overcards or the like. Just because your hand improves doesn't mean it's improving to the best hand, and many of your outs may be "dirty". Avoid marginal calls.