Does rejection of null hypothesis (p<α) for one representative sample imply rejection in a different sample?

Definitely not.

Consider a hypothesis test carried out at level $\alpha$. When $H_0$ is true, you reject a fraction of the time that is $\alpha$ (or for composite null, no more than $\alpha$).

So imagine $H_0$ is true and you reject. You then get a new sample at random from the population. The probability that the next sample would result in rejection would still only be $\alpha$.

Now consider a situation where the null is false, but where the power is 50%. Then the probability that the next sample will reject given the current sample rejected is a toss-up.

In particular, for two random samples taken independently from the same population, the probability of rejection for each is normally independent of a rejection in the other (leaving aside some potential issues with finite samples, and so on).

Can I think of these 5 times as obtaining p values less than α?

Yes, the test statistic falling into the rejection region corresponds to a p-value $\leq \alpha$

Does rejection of null hypothesis (p<α) for one representative sample imply rejection in a different sample?

Definitely not.

Consider a hypothesis test carried out at level $\alpha$. When $H_0$ is true, you reject a fraction of the time that is $\alpha$ (or for composite null, no more than $\alpha$).

So imagine $H_0$ is true and you reject. You then get a new sample at random from the population. The probability that the next sample would result in rejection would still only be $\alpha$.

Now consider a situation where the null is false, but where the power is 50%. Then the probability that the next sample will reject given the current sample rejected is a toss-up.

In particular, for two random samples taken independently from the same population, the probability of rejection for each is normally independent of a rejection in the other (leaving aside some potential issues with small populations, and so on).

Can I think of these 5 times as obtaining p values less than α?

Yes, the test statistic falling into the rejection region corresponds to a p-value $\leq \alpha$

Does rejection of null hypothesis (p<α) for one representative sample imply rejection in a different sample?

Definitely not.

Consider a hypothesis test carried out at level $\alpha$. When $H_0$ is true, you reject a fraction of the time that is $\alpha$ (or for composite null, no more than $\alpha$).

So imagine $H_0$ is true and you reject. The probability that the next sample would result in rejection would be $1-\alpha$.

Now consider a situation where the null is false, but where the power is 50%. Then the probability that the next sample will reject given the current sample rejected is a toss-up.

In particular, for two random samples from the same population (with replacement between taking the two samples if the population is finite), the probability of rejection for each is normally independent of a rejection in the other.

Can I think of these 5 times as obtaining p values less than α?

Yes, the test statistic falling into the rejection region corresponds to a p-value $\leq \alpha$

Does rejection of null hypothesis (p<α) for one representative sample imply rejection in a different sample?

Definitely not.

Consider a hypothesis test carried out at level $\alpha$. When $H_0$ is true, you reject a fraction of the time that is $\alpha$ (or for composite null, no more than $\alpha$).

So imagine $H_0$ is true and you reject. You then get a new sample at random from the population. The probability that the next sample would result in rejection would still only be $\alpha$.

Now consider a situation where the null is false, but where the power is 50%. Then the probability that the next sample will reject given the current sample rejected is a toss-up.

In particular, for two random samples taken independently from the same population, the probability of rejection for each is normally independent of a rejection in the other (leaving aside some potential issues with finite samples, and so on).

Can I think of these 5 times as obtaining p values less than α?

Yes, the test statistic falling into the rejection region corresponds to a p-value $\leq \alpha$