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?


Here is an example showing that you
could easily mistake a sample of size $n=100$
from $\mathsf{Gamma}(\mathrm{shape}=50,\mathrm{rate}=5)$ as
a sample from $\mathsf{Norm}(\mu = 10, \sigma=\sqrt{2}).$
Use R to sample $n=100$ observations at random
from the gamma distribution above, and summarize
the sample:
set.seed(2022)
x = rgamma(100, 50, 5)
summary(x);  sd(x)
    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
   7.182   9.182   9.947   9.979  10.903  14.278 
[1] 1.301298  # sample SD

A Shapiro-Wilk normality test fails to reject
the null hypothesis that the population has a
normal distribution (P-value $0.46 > 0.05 = 5\%.)$
shapiro.test(x)

        Shapiro-Wilk normality test

data:  x
W = 0.98733, p-value = 0.4602

Also, a normal probability plot (Q-Q plot) of the 100 observations is very nearly linear, as plots of a normal distribution should be.
qqnorm(x); qqline(x, col="blue")


Moreover, (falsely) assuming normality, a 95%
t confidence interval based on the sample
is $(9.72,\, 10.24),$ which includes $\mu=10.$
t.test(x)$conf.int
[1]  9.720308 10.236719
attr(,"conf.level")
[1] 0.95

Also, a 95% chi-squared CI for the population variance is $(1.3,\, 2.3),$ which includes
$\sigma^2 = 2.$
99*var(x)/qchisq(c(.975,.025),99)
[1] 1.305417 2.285194

Finally, a histogram of the 100 observations
seems consistent with a sample from
$\mathsf{Norm}(\mu=10,\sigma=\sqrt{2}).$
hist(x, prob=T, col="skyblue2")
 curve(dnorm(x, 10, sqrt(2)), add=T, col="brown", lwd=2)
  curve(dgamma(x, 50, 5), add=T, lty="dotted")


The solid brown curve is the density function
of $\mathsf{Norm}(\mu=10,\sigma=\sqrt{2}).$
Even if someone suggested that the population
might be gamma distributed, the density function (dotted) of $\mathsf{Gamma}(50,5)$ does not seem an obviously better fit
to the histogram.
Note: The inability to distinguish between
$\mathsf{Norm}(10,\sqrt{2})$ and $\mathsf{Gamma}(50,5)$ does not detract
from the practical use of either distribution in applied probability modeling. Normal distributions are familiar and often used. However, on theoretical grounds it may be preferable to use a corresponding gamma distribution in practice--especially if negative values are
impossible or if one needs to accommodate to occasional high outliers.
