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Hard to tell for sure without a close look, but the roofing / waterproofing material looks like TPO, a single-ply roofing membrance. It is not made for direct traffic, so appears someone has installed rubber pavers over it to protect the membrane. Not an unusual application. The membrane should have positive drainage to what appears to be scuppers, and any water falling on that balcony wants to be able to freely drain - meaning, any pavers installed should have a means of drainage on the underside. Appears to me that these rubber walkpads really do not.
Have a professional roofer /waterproofing company take a look and work with you toward a solution.
That is Oriented Strand Board (OSB), typically used for sheathing. Looks like it has been used exposed to weather, which is not its intended use. There is no reasonable repair for that, other than replacement... with materials suitable for whatever the application might be. Call a professional contractor.
That is going to vary depending on where you live, the particulars of your home construction, as well as particulars of your planned "pop-top". Here in Richmond, VA, the structure and exteriors, plus a window or two for a relatively narrow dormer runs in the $10-12,000 range... before you make interior renovations to the bathroom which presumably you may be seeking.
I am guessing that you may be interested in something wide than a 3-4 foot typical dormer. Well, things become much more structurally complicated in a Cape roof, the wider this pop-top becomes. As the roof loads become asymmetrical, beams and opposing side structural work will likely become involved (ie: you may have to add structure on the OTHER side of the house, and a beam at the peak of the roof... possibly posting-down to the foundation, to balance the load). I have seen those go to $30,000+... again, before you start adding in the bath renovations below. I know; sounds shocking but in order to maintain the structural integrity of your roof, these things may be neccessary.
Not much info here to make a determination. Drainage issues can be tricky. Depends on what the builder was engaged to do (was the contract specific about installation of a certain system, or was it a performance specification?). If the contractor is a member of a local chapter of a trade association, such as the NAHB or NARI, they typically have ethics panels who can look at the issue and help you determine what you should have expected of your contractor. Likewise, if your state has a Board for Contractors.
So many people today resort to the threat of negative online reviews, but please be careful how you use those. That may be unfair to the contractor if there are contractual stipulations or specs he was to follow, and he may have done as contractually required, so trashing him online may not be warranted and can unjustly ruin the well-meaning contractor's means of livelihood.
You may also discuss your issue by engaging a legal professional. Again, hopefully that professional can review the agreement and determine whether there is cause for pursuit - what obligations the contractor had to you, and whether he has met those or not.
As committed industry professionals, we want to know that our peers are dealing fairly with their clients. I hope that you can come to understand rights and obligations in this case and resolve reasonably to your satisfaction.
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