(+1) for indeed a crucial, in my opinion, question.

In macro-econometrics you usually have much smaller sample sizes than in micro, financial or sociological experiments. A researcher feels quite well when on can provide at least feasible estimations. My personal least possible rule of thumb is $4\cdot m$ ($4$ degrees of freedom on one estimated parameter). In other applied fields of studies you usually are more lucky with data (if it is not too expensive, just collect more data points) and you may ask what is the optimal size of a sample (not just minimum value for such). The latter issue comes from the fact that more low quality (noisy) data is not better than smaller sample of high quality ones.

Most of the sample sizes are linked to the power of tests for the hypothesis you are going to test after you fit the multiple regression model.

There is a nice calculator that could be useful for multiple regression models and some formula behind the scenes. I think such a-priory calculator could be easily applied by non-statistician.

Probably K.Keeley and S.E.Maxwell article may be useful to answer the other questions, but I need more time first to study the problem.

(+1) for indeed a crucial, in my opinion, question.

In macro-econometrics you usually have much smaller sample sizes than in micro, financial or sociological experiments. A researcher feels quite well when on can provide at least feasible estimations. My personal least possible rule of thumb is $4\cdot m$ ($4$ degrees of freedom on one estimated parameter). In other applied fields of studies you usually are more lucky with data (if it is not too expensive, just collect more data points) and you may ask what is the optimal size of a sample (not just minimum value for such). The latter issue comes from the fact that more low quality (noisy) data is not better than smaller sample of high quality ones.

Most of the sample sizes are linked to the power of tests for the hypothesis you are going to test after you fit the multiple regression model.

There is a nice calculator that could be useful for multiple regression models and some formula behind the scenes. I think such a-priory calculator could be easily applied by non-statistician.

Probably K.Kelley and S.E.Maxwell article may be useful to answer the other questions, but I need more time first to study the problem.