Question: I want to start deep frying at home in a pot. I read that you can reuse the oil a couple of times and that the quality and taste decreases because of the bits at the bottom, that are being burned (basically sauted, right?) while the rest is frying.

I was thinking of using a pasta sieve inside my pot, which (hopefully) catches those bits, keeping them from burning at the bottom.

Has anyone tried that and does that sound like a good idea?

Answer: 
Your pasta sieve will not be fine enough and it will not prevent what you think it will. The particulates in the oil that burn are burning because your oil it hot enough to burn them. Keeping them suspended above the bottom won't prevent the burning. They will still be submerged in hot oil.

When your oil starts to get older simply run it through a very fine mesh or filter (I use coffee filters) to strain the sediment.
Yes, you can although you will need to line the strainer with something to act as a filter. We do a double lining of kitchen paper and when the oil is cooled, we run it through, jar it and refrigerate. Paper coffee filters will work well too.