1) Is it the investors' responsibility to report their LTCG and STCG to the IT department? Will they know it if we do not report our gains? (I am not planning a tax evasion, just want to know)
Yes it is the investors' responsibility to correctly report all their income from various sources. In case of gains in Mutual Funds, since the PAN is linked to our investments, it is possible for them to know how much gains you had vs how much you reported. Whether they choose to follow up on it is up to them (and they can do it retrospectively as well).
2) If without the help of platforms like Goalwise, how can the investor know how much amount pertains to LTCG and STCG in a SIP scenario where funds investment dates varies?
The respective Mutual Fund companies that you are investing with will provide a capital gains reports for your investments with them individually.
3) If an investor is investing and redeeming funds like liquid funds and ultra short term debt funds several times during a financial year, will it be necessary for him to remember all these transactions and gains pertaining to them for income tax purposes?
Same as above - you should be able to get a Capital Gains report from the respective MFs that you are investing in. You will have to consolidate them though.
We have written a blog about it as well which you might find useful - https://www.goalwise.com/blog/how-to-get-capital-gains-statement-for-your-mutual-fund-investments/