The simple answer is you might be able to, but shouldn't. First you need to be clear about what it is you are after - creating a 'room' (ie: a habitable space) is different to just enclosing and under verandah area to use a a casual workshop or storage area. The former would need plans and approval, the latter might, but you could probably get away without. In either case you want it fully waterproof - and that will require a new cover of some sort which might be able to be on top of the existing - or you might create a new 'roof' underneath (the 'false ceiling you refer too?). Not an easy job for a novice DIYer as you seem to be - and easy to get it wrong (especially on waterproofing).
Advice from me on this forum is general and for guidance based on information given by the member posing the question. Not to be used in place of professional advice from people appropriately qualified in the relevant field. All structural work must be approved and constructed to the BCA or other relevant standards by suitably licensed persons. The person doing the work and reading my advice accepts responsibility for ensuring the work done accords with the applicable law.