Regarding your first concern,
I agree with you that you have to control and make sure that the weekly update percentages represent the actual work on site. The easy solution for this problem is to have site audits.
Each week pick some activities and check their status by yourself to know if the percentage that the contractor submits to you is correct or just misleading data.
I think that the client must have an eye on the site team and check these update percentages periodically to avoid any future conflicts.
If you find any misleading data or wrong percentages between the documents and the actual work on the site, make sure to document this and open the topic in meetings to get the top management’s attention to this problem.