I'm trying to learn statistics myself, so this answer will be far from complete, but might help point you in the right direction until (hopefully) somebody more knowledgable answers.

The standard error of each parameter, like the standard error of the sample mean, is an estimate of the standard deviation of its sampling distribution. In the case of the standard error of the parameter the calculation is not so simple as dividing by the square root of the sample size. Equations (22) and (23) of this link give the formulae for the intercept and slope parameters in simple linear regression.

For propagating the uncertainty using Around, I think that the correct way to do it would be to extract the parameter confidence interval from your fitted model, which should be a list of two values {confMin, confMax} for each parameter. The standard errors for the parameters follow a Student's T distribution, so the confidence intervals are symmetrical, therefore this should give you the parameter with uncertainty:

Around[Mean@{confMin,confMax},(confMax-confMin)/2]

As an aside, if anyone from Wolfram Research is reading, the above post touches on a pain point in using Mathematica for statistics. It can at times be very difficult to discover the mathematical definition of named statistics (e.g. the parameter's standard error) from the documentation. Sometimes this is compounded by a particular named statistic having various different formulae associated with it depending upon the textbook or software package used, leaving the user unsure of which implementation Wolfram is using. This contrasts to R where the source code can be referred to.

It would be very helpful for the long term if Wolfram could, for example, make each formula it uses explicitly available by linking from a super-function's documentation to a page in a formula repository where the formula would be clearly stated in both mathematical language and Wolfram code.