Hi Niti Shah,
Marketo has a page that gives all of this information: https://www.marketo.com/education/marketo-certification/marketo-certified-expert/
and there are multiple threads on community that talk about what you need to pass. I recommend searching community before posting
30 day course to get Marketo certified New Marketo Certification Exam
How long to study for (is one month enough)?
It depends on your experience in Marketo. I've been using it ~4 years so I studied for a few weeks before my exam and passed. Marketo recommends 800-1000 hours of hands-on experience to be prepared for the exam. I'd qualify that by saying that the experience needs to be admin heavy, since there are questions about the admin section which not all users - not even all admins - use regularly.
When should one take the exam?
When you're ready 
but aside from that Marketo has University Day before Summit. You can sign up for classes to help supplement your knowledge in time for the exam and you can take the test on site. This is ideal for some people because if you're like me, the nearest testing center is 100 miles away and it's not always easy to do a proctored test at home.
Practice questions /Are there practice exercises?
There are practice questions online: Marketo Certification Preparation: Sample Questions
And there's a User Group you can join to help with certification: Certification Study Group
How to sign up?
Create an account here and you can sign up for the exam: Welcome to Marketo Exam Registration
Are professional courses needed?
There are study courses you can take, particularly at University Day but also from Marketo University and the Prep Course that Marketo offers: Marketo Certification: Expert Exam Prep Course
I would say that it depends on you as an individual. What is your experience level in Marketo? If you have less but want or need your MCE, a class may help you. If you have problems concentrating or motivating yourself to study, a class may help as well. That being said, many people take the class without having taken a professional course.
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