study permit

1) Can I apply study permit with in canada , if not what will be the procedure

Yes, you can apply for a study permit within Canada. To be eligible you must prove that you have enough money to pay for your tuition fees, living expenses for yourself and family members who come with you to Canada and return transportation for yourself and family members who come with you to Canada.

You must be a law-abiding citizen with no criminal record and not be a risk to the security of Canada. You may have to provide a police certificate.
You must be in good health and willing to complete a medical examination, if necessary.
You must satisfy an immigration officer that you will leave Canada at the end of your authorized stay.

2) If I decide to enroll at a "private career college" in Ontario as a full-time student, will I be eligible to work off campus?

Your study permit and the fact that you are enrolled full-time in a professional training program at a designated learning institution gives you the right to work off campus without a work permit. However, this will be indicated on your study permit. If your study permit indicates that you are allowed to work off campus, you are allowed to work for up to 20 hours per week while your program is in session and full-time during school breaks.

3) Also, would my graduation from a "above designated private career college" satisfy the requirements for me to subsequently apply for a "Post Graduate Work Permit"? If it is what will the period

You would be able to apply for a post graduate work permit if you have a valid study permit when applying for the work permit, if you continuously studied full time in Canada and have completed a program of study that is at least eight months and if you have completed and passed the program of study and received a written notification from the educational institution indicating that you are eligible to obtain a degree, diploma or certificate.
You must apply within 90 days of receiving written confirmation from the educational institution indicating that you have met the requirements for completing their program of study.

5) If my spouse accompanies me throughout the duration of my studies, will he be allowed to work in Canada? (If so, how many hours per week?)

He needs to obtain a work permit in order to work in Canada.

6) Lastly, my two kids(16 year and 8 year old) will also accompany me throughout the duration of my studies in Canada. Is it necessary for me to apply for a study permit for them ? Also, whether Canadian government will be provided the education for my kids free of charge or do I have to pay for their education too,

Your children must have study permits in order to go to school to Canada. In Canada we have public schools and private schools, the government pays part of the fees for public schools, therefore you will need to pay the other part of the fees. If you decide to enroll your children in a private school, you will have to cover the totality of the fees.