When the DM is largely responsible for giving the party magic items (as is the case with 5e), it is absolutely the fault of the DM for throwing something against them that multiple members of the party are fully useless against, especially at a low level.
I absolutely agree with you on the Stunning Strike matter. The nerf has helped - it's gone from "Use almost all my Ki on Stunning Strike." to "What can I afford to use Ki on while Stunning Strike is on cooldown?" But it's still the wrong kind of question to be asking, and IMO it's exactly the kind of debilitating ability that makes DnD unfun.
I'm not incredibly experienced, but for the first question, AoN cites Core Rulebook pg. 592 as:
If a staff contains a cantrip, you can cast that cantrip using the staff without expending any charges. The cantrip’s level is heightened to the same level as cantrips you cast.
So I don't believe a cantrip requires charges to cast, even from a Staff Nexus staff.
And I'm also not seeing any contrary guidance on the unchangeability of the Staff Nexus spells, which quite frankly is a little odd to me, given the rest of the system's more flexible approach.