Is it permissible to count the amount given as a loan to a friend experiencing financial hardship from zakat al-mal (money zakat), or to give zakat for this amount? And is zakat obligatory on money invested in gold trading with a friend, with an agreement to take the profit annually?

It is not permissible to consider what was given to your friend as Zakat debt, as it was not given with the intention of Zakat, and deeds are judged by intentions. Zakat is obligatory on the money you gave your friend, whether it was a loan or a Mudarabah (partnership), because it is still your property. If it was a Mudarabah, Zakat is also obligatory on your share of the profit, if any. As for a loan, it is not permissible to take profit from it.