You're right. The problem of multiple comparisons exists everywhere, but, because of the way it's typically taught, people only think it pertains to comparing many groups against each other via a whole bunch of $t$-tests. In reality, there are many examples where the problem of multiple comparisons exists, but where it doesn't look like lots of pairwise comparisons; for example, if you have a lot of continuous variables and you wonder if any are correlated, you will have a multiple comparisons problem (see here: Look and you shall find a correlation).
Another example is the one you raise. If you were to run a multiple regression with 20 variables, and you used $\alpha=.05$ as your threshold, you would expect one of your variables to be 'significant' by chance alone, even if all nulls were true. The problem of multiple comparisons simply comes from the mathematics of running lots of analyses. If all null hypotheses were true and the variables were perfectly uncorrelated, the probability of not falsely rejecting any true null would be $1-(1-\alpha)^p$ (e.g., with $p=5$, this is $.23$).
The first strategy to mitigate against this is to conduct a simultaneous test of your model. If you are fitting an OLS regression, most software will give you a global $F$-test as a default part of your output. If you are running a generalized linear model, most software will give you an analogous global likelihood ratio test. This test will give you some protection against type I error inflation due to the problem of multiple comparisons (cf., my answer here: Significance of coefficients in linear regression: significant t-test vs non-significant F-statistic). A similar case is when you have a categorical variable that is represented with several dummy codes; you wouldn't want to interpret those $t$-tests, but would drop all dummy codes and perform a nested model test instead.
Another possible strategy is to use an alpha adjustment procedure, like the Bonferroni correction. You should realize that doing this will reduce your power as well as reducing your familywise type I error rate. Whether this tradeoff is worthwhile is a judgment call for you to make. (FWIW, I don't typically use alpha corrections in multiple regression.)
Regarding the issue of using $p$-values to do model selection, I think this is a really bad idea. I would not move from a model with 5 variables to one with only 2 because the others were 'non-significant'. When people do this, they bias their model. It may help you to read my answer here: algorithms for automatic model selection to understand this better.
Regarding your update, I would not suggest you assess univariate correlations first so as to decide which variables to use in the final multiple regression model. Doing this will lead to problems with endogeneity unless the variables are perfectly uncorrelated with each other. I discussed this issue in my answer here: Estimating $b_1x_1+b_2x_2$ instead of $b_1x_1+b_2x_2+b_3x_3$.
With regard to the question of how to handle analyses with different dependent variables, whether you'd want to use some sort of adjustment is based on how you see the analyses relative to each other. The traditional idea is to determine whether they are meaningfully considered to be a 'family'. This is discussed here: What might be a clear, practical definition for a "family of hypotheses"? You might also want to read this thread: Methods to predict multiple dependent variables.
This depends on what question you are trying to answer and what your strategy is.
I like to think about what would happen to my conclusions if I were to add to my data some additional columns of randomly generated noise. In your case this would add more correlations.
If you will declare success/significance if any of the correlations are significant (fishing for significance) then yes, you need to do a correction for multiple comparisons because if the truth is nothing is correlated, but you add a bunch of random noise variables and don't adjust, then you will likely see something significant by chance.
If on the other hand there are specific comparisons that are of interest and would have been of interest if only those 2 variables had been in the study/dataset, then you probably don't want to adjust for multiple comparisons. Think about a case where 2 variables are correlated, but you would prefer them not to be (what I want to eat, but my wife doesn't want me to eat vs. a measure of my health), you could add a bunch of random noise variables and adjust for multiple comparisons and the adjustment would change a significant result into a non-significant result (great for justifying my snack, but not really honest).
Best Answer
Firstly, when you perform multiple hypothesis tests (as you do by looking at whether p-values from multiple outcomes), there is in principle a multiplicity issue in the sense that comparing each p-value versus the level $\alpha$ will result in a familywise type I error rate $>\alpha$. I do not think this really goes away, if you do cross-validation. Whether you need to control the familywise type I error rate and across which analyses is a complicated issue. E.g. if you write two separate papers on the same dataset, do you get twice the familywise type I error rate, but not if you put the results in the same paper? This is really only (relatively) clear in a few settings such as confirmatory clinical trials for getting regulatory approval for a drug.
Secondly, the practical reason why many people are keen on adjustments is, because many people take a "many shots on target" approach, where they do lots of comparisons and then emphasize those with an unadjusted p-value <0.05. It is clear that when people study a lot of things including a huge number of things that really do not affect the outcomes being studied, that this will fill the scientific literature with many purported findings that are just random noise. This only gets worse when there are many small decisions left open until the data has been collected, which may lead to the potential for the choices in the analyses being data dependent. It may be debatable whether multiplicity adjustments help that much for such issues, but I guess I am not alone in trusting results with p<0.05 less when (a) the study was not pre-registered with outcomes and analyses pre-specified, (b) lots of outcomes were studied, (c) no adjustment was made for multiplicity and of course (d) the claimed effect is not a-priori plausible.
Thirdly, you should not completely change your view of results just because p=0.02 or p is just over 0.05. The former is not completely compelling evidence (and I would not get too excited about it) and the latter does not mean that the hypothesized effect is not there. Of course, this may affect what journal editors and reviewers will let you write (and whether they get excited about your paper) so in practical terms it may sadly be a major difference.