Null Hypothesis

Q1:Does the null hypothesis apply on the burden of proof?

I'm guessing you mean something like this:

H-0: The effect of X is not significant

H-1: The effect of X is significant

"significant" can be switched with 'exists', in the case of a debate regarding whether something is real:

H-0: X does not exist in reality

H-1: X does exist in reality

It is unclear what you mean by 'apply' in this case. Do you mean, "the null hypothesis is the default until evidence can be shown? If so, yes but only in the sense that the person has failed to prove their claim. You can't say "you can't prove it's real, therefore it is not real", as this is an Argument from Ignorance. However, for extraordinary claims that would invalidate some accepted logical truths or empirical knowledge, it is good to practice skepticism, and so certain claims can be rejected if they are more likely to be false than true (as long as there is no evidence for them).

Q2:Can we use it to determine whether an analogy is correct or wrong?

Analogies are analysed simply by looking more closely at the comparatives, or the things that are said to be analogous (basically, what you're comparing). Often the line of comparison (the commonality being looked at in both comparatives) can be challenged. If it is too dissimilar in the comparatives, then the analogy is said to be weak.

Logical form:

X is like Y.

Y has property P.

Therefore X has property P.

(But X is unlike Y).

Basically, the properties you're claiming the comparatives share actually need to be present, and strongly so, in both comparatives. If not, then the analogy is weak.

I do not believe this applies to the logical fallacy issue, because of the methodology. A null hypothesis says that "I believe there is a difference between X and Y. I will test the null hypothesis that there is no difference, and if I fail I will conclude there is a difference".

But I am not saying if X not = Y then X = Y. The null hypothesis is proposed in a statistical test, not an argument of fact. If I do this test 100 times, and 95 times there is no difference, than I conclude that there is no significant difference. If (under the p<=.05 standard) I get a difference in more than 5 cases, I will conclude that there is enough difference to be significant. But that does not mean every time you put X and Y together they will be different. A test like this does not make a 100% argument, it does not produce a false dilemma. It is a probability.

In tests that talk about null hypotheses, challenging the test is not based on errors in logic. The errors occur in the methodology, or the data used, or the testing criteria and assumptions. But the logic has been vetted for centuries.