Adjusted P_values I was wondering if we need to use adjusted P-values when we use several multiple regressions (same independent variables and covariates, but different dependent variables)? Also in multiple comparisons test, (bonferroni/Tukey) are based on adjusted p-value or we need to calculate it by ourselves?
 A: I'm not sure I'm going to answer you question exactly, but I wanted to clarify what an adjusted p-value is and why we use it.
The limit $\alpha$ you set for the p-value is the probability to find a false positive : the probability to reject the null hypothesis when the null hypothesis actually holds. The equivalent in regression is the probability that the observed correlation is due to random (non significant).
When performing several tests, the probability to have at least one false positive in all the tests is of course higher than in a single test. 
Take a simple example. You want to test if a coin is unbiased (unbiased = null hypotheses). One test consists in 100 heads or tails and you measure the number of heads. Using $\alpha=0.05$, you get the confidence interval $[40;60]$. The test is simply :


*

*if you get between 40 and 60 heads you accept the null hypothesis : unbiased

*otherwise you reject it : biased


Now you run this test ten times. If the coin is unbiased, the probability to reject "unbiased" at least once is :
$\alpha_{10}=1-(1-\alpha)^{10}\approx 0.4$. You get 40% probability to find at least a false positive. More generally the formula is for $n$ tests :
$$\alpha_n=1-(1-\alpha)^n$$
In science, a false positive is a false discovery. Not using adjustment can lead to false results easily. Imagine a researcher tests a hundred hypotheses and only publish the successful ones (null rejected). He doesn't even write anything about the failed ones : no one cares about inconclusive attemps. Then it looks like the probability to publish something false (the observation of a random deviation) is $\alpha=0.05$ while it is actually $\alpha_{100}=0.994$.
Bonferroni method : for $n$ tests, use $\alpha'=\frac{\alpha}{n}$ instead of $\alpha$ for each of the $n$ tests. See : https://onlinecourses.science.psu.edu/stat503/node/15
In our case, we can understand Bonferroni's ideas with basic maths :
$$\alpha'_n=1-\left(1-\frac{\alpha}{n}\right)^n\approx 1-\left(1-n\frac{\alpha}{n}\right)=\alpha$$
Using the Bonferroni adjusted $\alpha'$ for each test insures that the probability to have at least one false positive in all the tests is $\alpha$.
Of course multi-hypothesis are not $n$ identical hypotheses, and not always independent of each-other like in my heads or tails tests. And there are considerations Bonferroni's basic idea is too much conservative. That's why the real methods are more sophisticated. But the underlying motivation is the same. I think this article is quite good : https://en.wikipedia.org/wiki/Multiple_comparisons_problem
To answer your question : yes you must use adjusted p-values when doing more than a very few regression/tests, especially if you don't publish the inconclusive ones.
A: As long as you are fitting multiple models and having more than one hypothesis test, you should carried out multiple-testing correction. The p-value obtain in your model is based on a single hypothesis testing. When you having multiple hypothesis, it is important to carry out the correction to adjust the p-value accounting for multiple hypothesis being considered simultaneously.
Tukey, Scheffé and Bonferroni are different methods which performs multiple testing correction on p-values. If you already used the method, then the p-value obtained are the adjusted values and no further calculation would be required but you can continue to carry out post-hoc tests in the case where there are group comparisons like in ANOVA. 
There are also other adjustment methods such as Benjamini-Hochberg's method which are used for controlling overall false-discovery rates.
A: As long as my understanding is right, 'adjusted p-values' are a set of methods for multiple comparison including Bonferroni/Tukey. Tukey is  used only to compare pairwise mean in ANOVA. So I guess that you should only use it when your regression has equivalent structure. 
Calculating p-values based on Bonferroni is just original pvalue/number of comparisons. For Tukey, You can use function TukeyHSD in package 'stats' in R 
Multiple comparison is the last thing you should worry about in a linear regression. I would suggest checking model selection/functional forms/residual analysis, etc and make sure that everything is right. 
