Anyone can apply OPT after 9 months being as full-time student. No question about this.
CPT has to be full-time student and the job is course related training or project related with thesis. No question about this too. A lot of schools does not sign CPT easily.
For applying H-1: you need have US degree, right? or you can have experience to compensate the US degree (documents required to support this). If advanced H-1B cap is needed, US advanced degree is required.