| | Re: Passive in the sentence ''I would like you to be doing something''
I'd disagree with bread_baker. For me, ''I would like someone to be being helped by you'' is grammatically correct - the form is perfectly logical. You have the base proposition Someone is being helped ... preceded by I would like - which needs the following verb to be in the infinitive with to. So is in the base idea converts to be, giving I would like someone to be being helped ...
But as bread_baker and the others have pointed out, we'd probably never say it. It sounds clumsy - too contorted. Not everything which is grammatically possible in a language is actually said. So the problem is one not of form but of use.
Then, to clarify your confusion with despite. The gerund is used there because despite is a preposition - and verbs following prepositions are always in the ing form - think for example of Thank you for helping , or I look forward to seeing you. So if the base idea was It was done, then if you insert despite, it becomes Despite being done, with was converting to being. This is therefore quite different from the sentence that you queried.