You contemplate a visit to the UK to meet with the faculty of several universities with a view to enrolling in their PhD programmes. A friend of your family is paying for your visit. You have bank statements that will appear to match a funds parking profile with a large deposit and no activity that demonstrates an on-going lifestyle in India. You have supporting documentation for the deposit(s), but haven't accessed them in a way that demonstrates they are actually yours. You are unemployed.
Based upon what you wrote, your application is going to be very weak. To start, we can look at the UKVI formulae that the ECO will think of right off the bat for your refusal letter (they have an inventory of standard text to draw upon)...
The onus is on you to qualify for entry clearance based upon your own
circumstances and your intentions. You have said that xxxxxx will help
you with this visit by paying for your maintenance and accommodation
whild you are in the United Kingdom and I am satisfied that he is in a
position to do so. But whilst I take that into account in assessing
your proposed maintenance and accommodation in the UK that is one
aspect of the visitor rules and this sponsorhip does not satisfy me of
your own intent to leave the UK on completion of your visit.
The key words here are "your own circumstances". What this means is that sponsorship satisfies one of the rules, i.e., that you will not go hungry for those 10 days. But there are other things to prove also and these things are proven by looking at your personal circumstances.
As icing on the cake, they might include another one of their standard formulae along this line...
...it is therefore unclear why your sponsor is providing you
accommodation and living expenses in the UK. Furthermore you have not
submitted any evidence to show they are able to provide this support.
Given the above, I am not satisfied you have a genuine professional or
personal relationship with your sponsor or that your sponsor can
provide support to you for the intended duration of your stay...
When somebody like a friend offers to pay for everything, that's wonderful, but the notion of 'coherence' comes in to play, it needs to make sense in the bigger picture. There are questions like why are they doing that for you? Do they do it for everyone? Do they have a history of sponsorship that UKVI knows about such that the ECO will trust (or distrust) them? Can they actually afford to take on the burden?
Honestly, when somebody just pops up out of the blue offering to pay for everything, UKVI will be justifyably suspicious of a contrivance and this damages an applicant's credibility.
Coherence is really important and we see lots of refusals where people didn't sit down and consider if the whole arrangement made sense from end to end. Incoherent applications come from the complete range of applicants, including people with advanced degrees up to and including PLAB registrants. Avoid incoherence!
With these things in mind, we can turn to your questions...
my query is - though this trip is costing me nothing ( as being
sponsored by family friend), is it still mandatory to provide my
father's bank statements?
The only mandatory items are a passport, some photos, biometrics, the application itself, and the fee (Paragraphs 33 and 34).
The relevant question is what do you expect to prove by submitting your father's bank statements? You already have a 3rd party sponsoring you, and they will be confused about what you are trying to tell them. They already know there's a strong connection between you and your father (I assume so because someone must have paid for your graduate and undergraduate degrees), but that's not enough to keep you from going underground in the UK if you are so minded. His bank statements can show that he is established and has a stable lifestyle, but this does not provide information about YOU and your circumstances. Incoherent. His bank statements are unlikely to improve your application, but nothing prevents you from doing putting a few in.
my personal bank account has above mentioned balance for last 2 months
( I have sufficient proof of where the money came from) but before
that it was pretty much inactive and low balance of about few hundred
pounds, will that be a problem?
Yes, it is likely to be a problem. Incoherence again. You have an account with GBP 17k and yet you need a sponsor to support you 100% for 10 days. The ECO is going to ask "What is that about?" This doesn't make sense. Moreover the account is a picture perfect image of funds parking, i.e., it does not show that you really have access to those funds and you are working against your own credibility by relying on a sponsor. In addition to capacity, bank statements are used to show the person has social and economic ties to their country.
Being sponsored for the trip, still need to show 6 months bank
This is answered exhaustively over here: Should I submit bank statements when applying for a UK Visa? What do they say about me?
So to bring this full circle, you are going to need a compellingly strong premise in order to overcome the weaknesses in your evidence. It means there's stationery and letterhead from the various universities in a formal communication that expresses interest in speaking with you personally. It should be written minimally by someone at the Associate Professor level that they can Google and call on the phone if necessary. You would also need a closely hewn itinerary that supports your premise. Against these needs, the banking evidence has a secondary priority.