I will be travelling from Minneapolis to Chicago on a domestic flight to catch an international flight to India. What rule will be applied for my baggage from Minneapolis to Chicago?
That depends if this is a single ticket or not.
When you have a single ticket covering multiple airlines or a combination of flights with different rules, such as domestic and international as per your example, there is a Most Significant Carrier rule which says which one rule for checked baggage applies. Usually that will make the rule of the international flight apply but the actual agreement (IATA Resolution 302) is not that simple. Cathay Pacific has their own page which briefly explains how the MSC rule works (at least from their perspective). This is just one example, so you should check with the actual airline(s) you are flying for their version.
When you have two tickets, then you have two tickets each with their own rule. The bookings do not know about each other, so it makes no sense to expect one booking to affect the other.
There is no such rule for cabin luggage, including carry-on and personal items, so for those you must comply with the smallest and lightest allowance of all your flights.
There are many variables that can affect this, but assuming both flights are with the same airline and are booked on a single ticket, checked baggage allowance and fees will typically be based on your international flight (i.e. calculated as though you are flying directly from Minneapolis to Mumbai and not on two separate flights). This is also likely to be true if the international flight is with a different airline with which your first airline has a codeshare agreement, but will likely then be based on your originating airline's luggage fees and rules.
If the bookings are on separate tickets, then you will pay bag fees on each flight, without regard to your other flight.
This can vary by airline, too, so it's best to look up your originating flight's airline's website to see specifically how your airline does this.
For luggage (weight) allowance, you can go by the international wheight allowance (usually 23 + 23kg or 23 + 32kg) as long as you've booked all the flights in a single reservation. If the companies are partners, the domestic one (usually just one 23kg piece) will let you check in your international luggage with no extra charge.
Regarding security checks and forbidden items, always go by the stricter rules. For instance, Germany has very strict rules against Zippo lighters, they almost apprehended mine once, luckily the guard said he would allow if I emptied out the lighter by removing the filling and wick, and if you put it in the checked luggage is even worse, they can open it and take it away. Denmark on the other hand has no problems with Zippos (even if filled). So my advice is always go by the stricter rules in this matter, or risk having your property apprehended.
In terms of pure luggage allowances, it depends solely on the airline. I flew on Delta this year, on a single ticket going from Seattle to Paris CDG. We had a stopover in Minneapolis. We had no issues with the luggage because it was all handled by Delta in Seattle and considered as an international flight for the whole stretch.
Rewind back to December of last year, we flew on Delta from Seattle to Tokyo - this time with a flight to LAX first (also single ticket, denoted as a stopover). We had no issue with the two checked bags policy as outlined in @Rafael's answer, but the flight from Seattle to LAX was treated as a domestic flight, where we were not allowed two carry on items and a personal (they did offer to check them for free, but nobody likes to have fancy electronics be checked - amirite). The LAX to Tokyo portion wasn't an issue because that was an international flight.
You should look on the airline's website or give them a call to find out.