Min Salary required in Monchengladbach for a family of 3

You asked two separate questions:
1. What will be my take-home pay on a salary of EUR3600/month?
2. What will be my living expenses?

My answers:
1. This depends on several personal factors you did not mention (and possibly don't know about yet). If I assume you're married with one kid, not member of a church,public health insurance and wife has no income, you'll get around EUR2500/month payout.
There are online calculators with which you can check other constellations, e.g. brutto-netto-rechner.info (of course all in German only).
2. How much money you need and whether you can save anything will depend largely on your lifestyle and spending choices. The possible range is very wide. But I'll try to give you some pointers:
The survival minimum (poverty line) is EUR725/month per person. You will be above that, but EUR2500/month for three is not far from it. Thus you cannot have expensive holiday trips and should send your kid to (free) German public schools (private or international schools cost a lot, but aren't necessarily better).
Further information on living costs are available at numbeo.com

Thanks beppi for the initial set of information you have provided. I have some more. Would be great if you can help me with them.
1. You arrived on EUR 2500/month. I believe you have deducted all the applicable taxes. Did you also deducted all the insurance (heath, nursing, unemloyment etc. whichever applicable) or will that be further deduction on top of 2500 ? The reason for asking this is I could not make out some of the fields in brutto-netto-rechner.info. Thus would be great if you can help here.
2. I am an Engineering graduate and overall IT experience of 9.5 years. I have been working on a Enterprise product from last 4 years. I will be working on the same product in the offered job. I can give more details about my profile if required. My question is whether as per German market standards for IT professionals the salary offered is less or ok ?
3. Does German companies give the one-time relocation charges for foreign nationals ? (We have something of this sort in India if an individual relocates from one city to another)
4. The Employer has asked me for my expectation about the salary. Apart from all the expenses I am looking for around 1000-1500 EURO as savings. So how much shall I quote as my expected salary.
5. With 3600 per month the salary is 43200 per year. I got to know for Blue Card you need to have salary above 50,800. If I get salary above this, then will I be applicable for Blue Card ? What all advantages I will get by having a Blue Card.

1. The calculation I made considered all compulsory deductions, with some assumptions (e.g. public health insurance, not private).
2. I think the offered salary is too low for your profile - but foreigners in their first job here are often offered less (because of the higher processing effort and risk you present to the company, and also because you will work less efficiently without good German language skills).
3. This is usually negotiable.
4. It is unrealistic to want to save that much. You should quote as much as you think you are worth to the employer.
5. You can read about the Blue Card online.