When globally distributing our software product, what do we do when someone somewhere says we are infringing their trademark/other IP?

First of all, register your trademarks in the countries you intend to market. Second, find some countries to host your service where you are definitely sure you are not infringing other IPs.

Patents and trademarks are entirely different.

To allegedly infringe someone else's patent, your product would have to include each and every feature claimed within one of the claims of the patent that you are alleged to infringe - if your product includes each and every feature in at least one of the claims of the patent, assuming the patent is valid, which is a whole other subject - you can have a professional conduct a validity study to see if they can knock out the patent or not - then you would infringe.
If the patent claim(s), claim any features that are not in your product, you will probably not infringe.

As for TMs, the basic test for TM infringement is the likelihood of confusion in the marketplace, in other words, is your product named/styled, whatever, to appear confusingly similar, or in the worst case, identical, to the TM in question. If so, you are likely to infringe. It also depends upon how strong the TM in question is - for example, even though APPLE markets computers and other electronic devices, if someone came out with a driverless car and put the APPLE logo on it, that would likely be infringement because APPLE is such a strong mark, and it is possible that APPLE might start marketing driverless cars as GOOGLE is also intending to do. If the mark in question is not necessarily that strong, then you can probably get away with using it or a similar mark, particularly in connection with different goods because the
"brand" protected by the mark is probably not that strong to transfer over or cross to other product areas.