Is it possible to simultaneously call multiple regression coefficients significant? Okay, this is a thought experiment:
Suppose you have a dataset with 40 covariates. Suppose the data has every nice theoretical property you could want: randomly sampled, variables not correlated, etc. Suppose it is also the case that the null is true for each beta coefficient at a 95% level AND we don't know that the null is true in all cases.
Next, you run the model, and 2 of the 40 betas appears to be significant, with a modestly large t value, but nothing crazy. Certainly enough to reject the null for both of these betas. Now, given that it was assumed the null is true, this is a perfectly reasonable result. Each beta has an approximately 5% chance of rejecting, so you would have expected two betas to reject before you even run the model!
This brings me to my issue: in what sense can we say multiple betas are significant when we ALWAYS expect a certain number to reject just due to randomness? AND how is it possible to conduct hypothesis tests for these betas simultaneously? Isn't that cheating, sort of like rolling a die until you get a six? It seems to me you can only pre-specify a single beta to look at before you run the model to get "true" significance.
To clarify what I mean by simultaneous: I am not talking about the overall test (F-Test, for example) or any related mitigating actions like multiple comparison corrections. I fully agree that these tests and measures are effective and make sense. I am asking at the level of the individual betas themselves with their individual CIs: if two betas reject, should I just ignore this and say: "Well, I expected two to reject. These effects are not real"? This may be my main point: does significance at the regression coefficient level mean anything in this case? Or suppose four had rejected. I would guess two would be false signals by chance, but how do I know which is real and which is fake?
PS: If you have an open-source text to cite and support your answer, I would love that.
 A: It's true that with 40 variables, you would expect two to be significant (when using the conventional alpha of .05) by chance alone.  That's why we shouldn't interpret the individual t-tests for a multiple regression model until after assessing the F-test for the model taken as a whole.  (It may help you to read my answer here: Significance contradiction in linear regression: significant t-test for a coefficient vs non-significant overall F-statistic.)  If I had a model with a non-significant F-test, but 2 out of 40 variables were individually significant, I would interpret those results as not actually meaningful.  
To be clear, I would not quite say, 'these effects are not real', because that isn't a valid interpretation of a non-significant result (see: Why do statisticians say a non-significant result means “you can't reject the null” as opposed to accepting the null hypothesis?)  I would say that you don't have sufficient evidence to reject the null.  
Regarding the follow-up question (how to know which 2 of 4 are real and which fake), you'll never know if significant variables have a "real" relationship with the response, and you'll never know if non-significant variables don't.  That's part of the nature of the game we're playing.  If you were really concerned about the possibility that some of your results might be false discoveries, you could use false discovery procedures to explore that, but it isn't common to do so in the kind of situation you describe.  
A: If I'm reading your question correctly you are simply asking about multiple comparison in statistics which is a well known phenomena. To remedy this, you can correct your significance using something like a Bonferroni-correction, although many types of correction methods exist.
Note that you also need to consider the implication of your analysis. Is it something that will dictate how your company or medical department operates? Then you should probably take it more seriously than if your study is simply to pave way and direct future research. Multiple comparison and how we address it, is in essence just a dance between type-1 and type-2 errors.
EDIT: After reading your edit.

This may be my main point: does significance at the regression
  coefficient level mean anything in this case?

The coefficient tells you both the direction and size of the impact your covariates have on the dependent variable, with respect to the model as a whole. If the p-value of a covariate is significant, but the model contains enough individual covariates with no pre-determined hypothesis, that you suspect the unadjusted p-values may be erroneous, the coefficient and the CI may be inflated, even if truly significant. Another important part with respect to coefficients is how well you have controlled for confounders as these can dramatically change the coefficients in your model. Related literature on this topic.
