I think that I can answer those questions.
I'm assuming, this is a submission-form, only for creating new tickets... Once the ticket is submitted via this form, how will the subsequent ticket thread appear... ?
submitting via this would create a ticket.
submitting again would not add it to an existing ticket and would open a new ticket.
Using osTicket's HTTP API, would it be possible to have a different ticket-submission form, based on the submitter's (end-user) organization ?
The form code does not appear to be setup to do that, but it is something that you could do, or you could setup different sites using this and brand it each one and then give the link to the respective company to use.
Using osTicket's HTTP API, would it be possible to have dynamic conditional field's like drop-downs, in the form ?
If you coded them yes. But you would need need to actual help topics in osTicket that the correspond to. ie they would look different in the form, than they do in osTicket.