You will need the LMIA before you can submit your Closed Work Permit application. Because LMIA is used for closed permits, you are entitled to work only for that particular company. During your work permit application you have to provide details, such as application numbers, regarding the LMIA of your employer.
Processing times are posted on the CIC website but really do vary. My IEC - Young Professionals work permit was approved within 2 days after I submitted my application. I submitted on Thursday night and got my approval the next Monday morning.