Is it possible to get the boarding pass for another flight at a different airport?

Is it possible? Yes, there's nothing stopping them other then airline or station specific rules. This means there's no way for anyone to answer specifically.

But, your best option is online check-in and either print your boarding pass or keep in on your phone.

That depends on the details and it's hard to determine up front. Your best route is to determine if on-line check in is possible for your second flight. If yes, than just do that.

If no, it's unlikely Airline B will allow you to check in at airport X. The main reason why online check in is sometimes not available, is that the airline wants to do a documentation check at the check in counter. The staff at airport X may not have the authority or training to check docs for travel to airport Z. They tend to be sticklers about the rules, since the potential fines and legal entanglements are quite bad for the airline.

Unfortunately these rules change rather frequently. I see different methods even on the same routes with the same carrier. Sometimes you can check in online and sometimes you can't. Sometimes they check docs at check in, sometimes at the gate and sometimes not at all.

Sure it is possible but not always. If airline B has a presence at airport X than it is highly likely they can do that unless the second flight is too far ahead. This happened to me once when I had a 10 hour flight followed by a 15 minute connection (originally longer but shorted due to late departure) and they could not produce the second boarding pass. They said if the flight was more than 6 hours away, the system would not print a pass but this might depend on the airline.

If Airline B does not a presence at airport X, you can still try at one of its partners if you know which the airline alliance. Sometimes they do it, sometimes not. Even some airlines that appear the same to us consumers can be different entities. I had a similar connection between two flights and I could not get a boarding pass for the second leg which was domestic upon checking in for the first international leg. I was told they were different companies and used different systems even though to me both were on the same carrier which they sold me over the phone by the same agent.

Online check-in is always a good idea. You can sometimes do it from the airport using WiFi or at an Internet Cafe or similar. Most online check-in allows people to check-in 23 hours or so in advance.

Depends on the airline and route.

In March 2016, I flew PTP-CDG on XL Airways, with an Air France connection to ZRH, and had checked in online for the connection. But nope, the Air France check-in kiosk at PTP would not recognise my booking, either through the reference number or me scanning my identity card, so I had to get the boarding pass at a kiosk located airside at CDG