It seems that there is already an answer on Cross-Validated that answers (partially) my question: Using non-stationary time series data in OLS regression.

To obtain useful results you can't use nonstationary data with OLS and time series, except in case of cointegrated series. There are other more advanced methods where nonstationarity is a non issue.

This does not mention the complete list of hypothesis that we would preferably check for such a regression.

But from answers by @Dave and @whuber I can infer that I'm free to validate or not the assumptions of Gauss-Markov theorem, and if doing so I can expect a BLUE just like for cross-section analysis.

However, from the responses given by Dave and whuber, it seems that the perspective underlying my question was wrong. I was under the impression that time series are a strange beast, in which assumptions for a valid regression were very special. This assumption seems generaly wrong, and time series regression should be considered very akin to cross-section regressions, except for the well-known case of time-series with integration order $I(1)$ or larger, which yield "spurious regression" unless those series are cointegrated.

@Dave and @whuber, thank you for your answers!