Is it permissible for a poor, sick person to take Zakat for treatment, then invest it in trade with the intention of securing enough for their treatment and basic needs, and is it permissible to give them Zakat while they are trading with that money?

It is permissible to give the needy from zakat according to their needs for treatment and other expenses. The zakat then becomes their property, and they can dispose of it to increase it and benefit from it. If their trade earnings are not sufficient for them, it is permissible to give them from zakat to the extent of their sufficiency; otherwise, it is not.