What should I check/test when I want to generalize from a sample to the population? How do I ensure that I'm allowed to generalize findings from a sample to the population? So far, I'm only aware of the standard error. Is this (alone/itself) sufficient or even valid?
As the CI is connected to that, do I need both of them respectively is the standard error meaningless without its CI?
Are there other tests, quantities and so on?


BruceET has a great answer.  I'd like to add some flavor based on a comment you've made:

I mean no matter how often you measure something, you never can be sure, then..?

Correct, this is just how inductive inference works.  I don't know that the sun will rise tomorrow, but it probably will.
Which leads me to my point.  We can't say anything for certain, but we can make probabilistic statements about phenomena. The Higgs boson is a particularly good example because it was accompanied by such a probabalistic statement (see $5\sigma$ as explained here https://blogs.scientificamerican.com/observations/five-sigmawhats-that/).  The p value, for all its faults, seeks to be such a statement: "Assuming all assumptions are met, and there is no bias in our estimation, there is a $p$ percent chance that we would see a test statistic at least as large were the null to be true."
Generally then, how can we generalize from sample to population?  That's a really hard job because we have to fight against a myriad of things, including but not limited to: sampling variability, internal and external validity, assumptions about the data generating mechanism which may or may not be appropriate, and so on.  Much of your confidence in making the generalization will not come from a single number such as the standard error, but will come from your contextual understanding of the measurement process, how the study was designed etc etc.  As I like to say "Statistics is not an algorithmic truth generating process" and so you'll never be able to test/compute something to tell you if you can generalize.
