There are no simple answers.  Everything is based upon the context it occurs in.

1. How do you ensure your requirements are of sufficient quality?

Firstly you need to understand what quality targets you are aiming for. Discuss this with your project manager and clients.

Now, an excellent way to ensure that you are writing good requirements is to define how they will be tested as part of the requirements statement.  What specific, measurable outcome will be achieved if this requirement is fulfilled?  What will happen if it is not fulfilled? (Take a look at this article for a more in depth discussion of this point.)

2. How do you ensure your requirements are valid?

The validity of a requirement is about how true it is. Again you need to understand that it's truth is in the context of the project;s goals.  So if you have a project implementing a payroll system you need to understand the true goal of the project. Is it better payroll processes, reducing the cost of managing payroll, or is it something else altogether.  

By understanding the goals you start to understand who's voices are important and who's voices are secondary.  And now I am talking about stakeholders and understanding their role on the project.  Which stakeholders are aligned with your project goals and which are along for the ride.  Which of them stand to gain the most from the project completing it's core goals on time?  ho is trying to fit in unrelated requirements into the project agenda?

Once you have this understood you know who you should be verifying requirements with.  This is another discussion for you and your project manager.