For H 0 </msub> : &#x03C0;<!-- π --> = 0.20 , for instance, the score test sta

Are there any theoretical considerations between p-value and the Bayesian statistics? I mean say, any theorem regarding both of these two concepts at a time.

Consider testing $H_0:\theta \ge 3\text{vs}{H}_a:\theta <3$ using a test in which the null hypothesis is rejected when the statistic $T=min\{X_1,...,X_n\}$, for example of size $n$ has small values. Compute the $p$-value using the sample $3.11,3.3,3.46$

We want to make an hypothesis test for the mean value $\mu$ of a normal population with known variance ${\sigma }^2=13456$, using a sample of size $n=100$ that has sample mean value equal to $562$.
Calculate the $p$-value.
Make the test with significance level $1\mathrm{\%}$ about if the population mean value from which the sample comes from is greater than $530$ using the $p$-value.
For the first one, about the $p$-value, do we have to calculate $P\left(\frac{530-562}{\frac{\sigma }{\sqrt{n}}}\right)$?
And for the second we have to check the $p$-value with the significance level, right?

Let $X_i$ be Gaussian random variables with $\mu$ = $10$ and ${\sigma }^2$ = $1$. We decide to use the test statistic $\hat{\mu }$ = $\frac{1}{20}\sum _{i=1}^{20}{X}_i$ and following tests:
$|\hat{\mu }-10|>\tau$: Reject $H_0$
$|\hat{\mu }-10|\le \tau$: Cannot reject $H_0$
1) Find $\tau$ if you want to have $95\mathrm{\%}$ confidence in the test.
2) You find that $\hat{\mu }=10.588$. Do you reject $H_0$? If so, what is the p-value?
Using textbook equations, I found out that for
1) $\tau$ = ${\sigma }_{\hat{\mu }}$ = $1.96\sqrt{\frac{1}{20}}$ $=0.392$
2) $|10.588-10|=0.588>0.392\to$ Reject $H_0$
p-value $=0.003$

Let $\mathcal{X}$ be a sample space, $T$ a test statistic and $G$ be a finite group of transformations (with M elements) from $\mathcal{X}$ onto itself. Under the null-hypothesis the distribution of the random variable $X$ is invariant under the transformations in $G$. Let
$\hat{p}=\frac{1}{M}\sum _{g\in G}{I}_{\{T(gX)\ge T(X)\}}.$
Show that $P(\hat{p}\le u)\le u$ for $0\le u\le 1$ under the null hypothesis

I'm confused about the interpretation of P value in hypothesis testing. I know that we set significance level as 0.05 which is the threshold we set for this test so that it won't suffer from Type I error by 5%.
And we are comparing P to significance level, does it mean P is the probability of making type I error based on the sample?

p-values behave uniformly. Now as p(np) is fixed and n goes to infinity, binomial converges to normal(Poisson). Now suppose I take random binomial samplings and fir normal(Poisson) to it, for say n = 1000. Will my p-value still be uniformly distributed or as binomial converges to normal(Poisson), p-values mostly will be in 0.8-1?