I'm confused because United Airlines only sent me a receipt for my purchase. I'm not sure of how I would obtain my boarding pass.
When the check-in window opens they'll send you an email with the link to check in. You can also go to their website with the reservation confirmation number (six-letters alphanumerical code) and find the "Check in" page. Since you have an Air India flight as your first leg, you may need to check in with them, and the reservation number may be different, so check the email carefully.
Will they not let me in the airport without a boarding pass? I would board at BLR, and I was thinking about getting the boarding pass at the airport itself instead of printing it.
You don't need boarding pass to go into the airport, and you can definitely print it there at the check-in counter or self service kiosks (if available). You do need the boarding pass to go through the airport security to the gates (what's called "airside").
Boarding passes are usually generated during the check-in process, within the 24-48 hours prior to the flight.
Would I go to the Air India counter or the United Airlines counter when I arrive at BLR? If I go to the Air India counter, would they be able to give me both of my boarding passes?
Air India, since that's who you're flying with. If both flights are on the same reservation, they will give you both the boarding passes, but you might still exchange it with the United-printed boarding pass in DEL (they'll tell you there, DEL, if you do, United does sometimes replace others' passes with their own on connections).
Would a US passport suffice for domestic travel in India?
I think so, I flew with my non-Indian passport within India with no problems.
When looking at the allowed luggage dimensions and weights, should I look at the ones for Air India or United Airlines?
It should be stated in the reservation confirmation that you got what the allowed baggage is. The dimensions and weights are usually the same across the airlines (but may differ between the ticket classes and your status - business class for example may sometime get higher weight allowances).
Is there a way I can find out the gate I will arrive at in advance?
Gates are usually assigned close to the flight, usually when you check-in the gate will be assigned - but not always, and it may be changed. When you're checking in the BLR gate may be assigned, but DEL may not yet be (since it's a later flight). Better check at the airport when you're there.
Is it okay if my carry-on bag does not close? I.e. it has a clasp, but it does not have a zipper and the inside contents are visible
Will stuff fall out of it during turbulence? As long as stuff doesn't fly out around the cabin when the plain shakes a bit you should be fine, but check with the gate agents to be sure. To be even more sure, bring a carry-on bag that does close securely.
Will my carry-on luggage need tags or should I add an identifying sticker?
Should? Yes. Need? No.
How can I access my boarding pass to print it out?
Depending where you're getting it. If you check-in at home, you'll get a PDF or a link in your email, if at the airport - they'll print it for you. Many airlines also allow mobile boarding passes (electronic QR code on your phone). Check with the airlines.
And don't worry, you've got this. Check-in agents and gate agents are there to help you if you are not sure what to do or where to go, and airport signs are usually very clear, and very similar in all the airports across the world.