Is the entire process of writing up an analysis done without prior consentment of a Moderator/QC member? If it is, then approximately how much time does a QC + GP approval take?
If you're planning on introducing a new set for a Pokemon, you don't need prior permission from anyone to put up a thread for it. However, you're not permitted to actually write out the comments in paragraph form, only in bullet points to make it easier for QC members to look through and approve/disapprove. If you get the 2 required QC approvals, you are then able to fill in the set's comments for the last QC member to give you full QC approval.
In terms of rewriting/updating a whole Pokemon's analysis, you have to look through each tier's respective analysis index thread and see which analysis needs updating and isn't taken by another user. If you plan on doing something that isn't seen in the index or want to update a Pokemon's analysis that is labelled completed, then you should ask a moderator/QC member before throwing up a thread for it. This is just to save you time in the event that the analysis is rejected for whatever reason when you went through the time to put it together.
QC + GP approval can take anywhere from 2-3 weeks in most cases, give or take a few days.
Extra question: how does one apply to be a QC member?
You don't actually apply for the team as far as I know. All you need to do is to consistently post quality and legitimate suggestions in analyses and get involved in discussions on IRC in the respective tier channels (#C&C is where OU takes place). You should probably be good at the tier you want to be a QC member for. Continue this for a good period of time and you should eventually get an invitation.