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For example, if I believe that dating methods are correct, why should I question their accuracy every time they're used. If they're right once, then they're right always.

I have answered that a couple of times before. If you can show that a widely used method is flawed, you will get a lot of attention, which means a lot of funding, awards, prizes, etc. Therefore scientists have lots of good reasons to disprove or at least alter existing theories. According to your belief Albert Einstein would have had no reason to write papers refuting Newtonian physics since Newtonian physics was so widely accepted in the early 1900s. Instead he did show the shortcomings, got his grants, jobs and Nobel prize. Or take a look at the more recent scandal concerning fake data by that Korean biologist. Within a few months an American scientist pointed out that the results are suspicious and have since been retracted. That's the self-correction inherent in scientific research.
Therefore claiming that methods which have been used successfully for several decades are simply wrong and that nobody has an interest in falsifying them shows a large amount of ignorance concerning science. That's why I was happy to hear that you will be studying biology in which case you should get a better understanding of how science (and the politics of science) works.

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Of course, recent discoveries have cast doubt on dating methods.

Again, you make a very bold claim but no specifics. Did you know that recent discoveries have shown the Old Testament to be a forgery ?

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My beliefs stem from the bible, which is fairly fuzzy on the subject of the age of the universe. However, I didn't say I thought the earth was 20,000 years old, I just don't think it could be much older than about that age.

So you are putting your gut feeling of maybe 6000 or 20000 against objective research that's been going on for 100 years.
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Frankly I wouldn't be surprised if the earth was about 6,000 years old.

The Babylonians that established their city-states 7000 years ago probably would be rather surprised by that.

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The 1980s eruption of Mt. St. Helens by example. Now, I'm not going to refute the typical evolutionist counter argument to this one just yet, because its laughably illogical and designed simply to steer the argument away from the problem.

You mean the response that a method which can only date material that's older than 2 million years will yield wrong results when used on a recent sample? Quite a surprise there I guess.
My bathroom scale has a disclaimer saying that it measures from 30 lbs to 250 lbs with an accuracy of +/- 0.1 lbs. When I put a 20 lbs weight on it it will indicate "0.0". Did I just disprove weighting technology or was I too stupid to read the disclaimer? Your pick.
http://home.austarnet.com.au/stear/mt_st_helens_dacite_kh.htm