I am an international student in the United Kingdom holding an Indian passport and I am planning to travel around Europe. Am I eligible to get a Schengen visa from the UK at the moment? Is Ireland included in the Schengen area? Is it a long process for getting the visa? And how long I can stay in each country?
You can certainly apply for a Schengen visa in the UK, as that seems to be your place of residence at the moment. Your stay under a visit visa would be a maximum of 90 days in any 180 day period, and this is counted for the Schengen area as a whole, so it doesn't matter how long you stay in an individual country, other than that you're supposed to apply at the embassy of the country which is the main reason for your trip (ie, where you spend the most time).
Ireland is not a member of the Schengen area, but instead forms the Common Travel Area (CTA) together with the UK. Your UK visa might allow you to travel to Ireland if it is issued under the BIVS (British Irish Visa Scheme) which is valid for Indian citizens, otherwise you would probably need a separate visa to visit Ireland.
As for the length of the process, I can't say too much. Whether you can enter and travel around specific countries varies a lot at the moment. You can refer to the page Reopen EU for details. Most likely, even if allowed to travel, you would need to quarantine for at least 5-14 days, so a vacation would not be very enjoyable. I don't know if this affects the visa process, but even if you manage to get a visa it doesn't mean you're guaranteed to be allowed to enter, if the current regulations forbid travel.