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1) Theoretically possible but practically no.  Sponsoring visas for parents is tricky as there are no established rules and entirely up to the discretion of immigration.  It's easier if you work for a big company, are at a managerial level and have a strong PRO.  In your case, I don't think it is likely or possible to do that i.e. your profession & salary

2) Usual documents for dependent visa are copies (scans) of first two pages of passport of child, photograph, birth/marriage certificate (attested by ministry of foreign affairs and bahrain embassy in your home country), mother's passport / residency etc. In your case, you will need to attach a divorce certificate (also attested as above).   

Read this:

If it is an online application by your employer i.e. through EMS system of LMRA; then originals are NOT needed.  But if you are a self sponsor or any category not handled by LMRA, then you would need to apply directly with immigration and they may ask to see originals.

Now the tricky bit; generally for a mother to sponsor kids here in Bahrain (and you must have a visa profession / industry allowing you to do so), they ask for an NOC from the father.  If the father is IN Bahrain on a work visa then this NOC actually has to be done at the Notary public and attested by Ministry of Foreign affairs Bahrain. If the father is not in Bahrain and you have a divorce certificate plus proof of custody, then it would not be needed.

My suggestion is to try and get an NOC from the father (have to attest it as well like the other documents - he will have to do it before a notary then MOFA India needs to attest it and finally the Bahrain embassy in India) or try to get any other official document which can serve as a substitute for NOC by proving custody in your favour.   

3) Who needs originals?  can you be more specific?   as I mentioned, originals are not needed depending on mode of application.  Even if they are; you can always courier them over via DHL - it takes 2-3 days max

If you get all the documents; it is theoretically possible to have everything done before 20th March.

Thank you