Alpha is the error rate of the binary significance test if at all. Not related to any of the observations.
However, since p is a loglikelihood ratio, it is related to the observation and null themselves.
To get actual likelihood you need to exponentiate, know the degrees of freedom and know one of the actual likelihoods. (Typically null is easier to come by as the other side is assumed to be null+treatment.) This exact likelihood is of course tiny. Which is why inequality is supposed to be used.
The process is not valid if any of the test's assumptions is violated in a big way.
The standard meaning of likelihood is P(data|Hypothesis)[1] where, for the purposes here, hypothesis will refer to "chance/accident generated the data". Do you agree with this? (I understand you say "kind of likelihood" because we are dealing with the inequality)
If so, can you clarify what you mean by "P is ... Likelihood of type 1 error (accidentally getting a result)..." ?
Actually I should have written L(parameters|data), I'm sorry for the confusion. Which is equal, for a given value of data and parameters, to P(data|parameters). But the likelihood function is not a function of the data (with the parameters fixed), it is a function of the parameters (with the data fixed). Its "meaning" is not a probability distribution of different outcomes given the hypothesis. But I maybe misinterpreted your comment.
A mathematical sleigh of hand depending on big N, chi-squared statistics and Wilks' theorem. This is obviously inaccurate for small N.
Otherwise p depends on the exact test employed. (Specifically the distributions employed. For Fischer's exact, hypergeometric, notoriously hard to calculate, but if you can, you can recover likelihoods too.)
However, since p is a loglikelihood ratio, it is related to the observation and null themselves. To get actual likelihood you need to exponentiate, know the degrees of freedom and know one of the actual likelihoods. (Typically null is easier to come by as the other side is assumed to be null+treatment.) This exact likelihood is of course tiny. Which is why inequality is supposed to be used.
The process is not valid if any of the test's assumptions is violated in a big way.
Other than this, all the points are true.