You can nail/secure your posts onto the rubber roof, it is done very frequently.
If you do this the posts will have to be flashed, ideally you will have a couple of the posts firmly secured to a wall, for bracing.
The problem with toe nailing your posts is that it is not a very secure connection in regards to lateral movement.
If your posts and railings move back and forth when subjected to normal use it can damage the flashings and create leaks.
So it is important to do it right the first time, there is a lot of work involved to correct the problem if it occurs.
If your roof slopes toward the house I can only assume that the water drains off of both sides and down your siding.
It is possible and even probable that the water is getting behind your siding extensively damaging your walls.
Termites, carpenter ants, and mold just love wet wood.
I can only assume that there is a door that accesses this deck, with the roof sloping towards the house it is vitally important that the bottom of the door be flashed.
You may need to remove the door to get the rubber under it.
You can use a termination bar and water cut off mastic for under the door but this is less than ideal for the long term.
How high is the bottom of the door above the roof deck?
What type of walking surface do you have over the rubber roof?
You can get a cheap digital camera for $100 and a memory card for $10 - $15.
If we had some pictures we can be of more help and offer suggestions as to other options available to you.
Otherwise we are just guessing based on your description and our own personal experiences with these types of roofs.
Is your roof similar to this
Or more like this

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