(This is, perhaps, a partial answer, to one aspect of the depicted situation.)
- We deploy.
Great! Good job getting a product out there!
- Then customers announce some additional requirements.
It sounds like those should have been part of the scope earlier.
Oh well, that's fine. Customers changing their mind is a thing that can happen. Their requests for something different should then be a new work request/project, or an adjustment to the existing project. Professional project management training specifies that adjustments to an existing project's scope should go through the "scope change procedure". That involves getting agreement with all stakeholders.
- Then we're hustling to meet those requirements. Devs, like myself, get urgent requests: "Quick! Develop this feature ASAP! We need to close the contract!"
Scheduling resources, like your time, is a separate topic than changing scope. I know, they both seem to impact the same resource (you) in the same way, and may be based on decisions by management, and so they do feel related. But internal scheduling is a separate skill that management (a "project manager", salesperson, and/or dev team lead/manager) may need to implement in some different ways.
(A scope change might be negotiated with a customer. Internal schedule probably would not be.)
- There's no time to develop it properly,
This is a resource issue.
Note tasks do not necessarily have a fixed scope, so providing an estimate is not really possible.
Improperly-scoped tasks can blow up, cause lots of stress, and make everybody involved less happy. All work should have a clear scope. For instance, a project's scope may consist of 14 tasks, and when each task is done, the project's scope is complete. Each of those tasks should be clearly scoped so that someone can verify whether those are done, or not.
If you don't have good scope, expect stress. Hpoefully you can convince whoever is relevant to improve scoping skill. Until that happens, as long as you keep working there, this type of stress will occur. There's no way you can work around that. The only way to improve that is... to scope better.
Note that while this answer does say there's quite a bit which is management's responsibility, many professionals with less experience may think they need more resources than they really do, and might prefer more time to be made available when management doesn't see that as necessary. And in at least some cases, management is right and the inexperienced professionals just need more experience showing how things are done successfully, and can then successfully deliver within tighter constraints (such as deadlines). So, regarding how much testing is really required, how much pre-planning is really required, what deadlines are actually realistic... it is management's job to determine these things. If they seem out of your reach, maybe the best solution is for your own skill to improve, or maybe it is to change employers so you can get into an organization with more reasonable demands. The truth may rely on a delicate balance which can be hard for an uninvolved unbiased and mostly uninformed person to really effectively take one side or the other.
But regarding things being unscoped, or scoped poorly (e.g. needing revision of clarifying completion without a good revision of resources like time), that's just plain bad. Maybe forgiveable, and hopefully improveable. But, still, bad.
Poor scopes just plain won't work out well, and will continue to not work out well until something significant changes, as you seem to have already noticed so far. Maybe try to have some conversations that demonstrate respect (regardless of how much respect actually exists in your inner feelings). See how you might be able to help improve. (e.g., provide clear data that can be used for making better scopes early on.) But don't expect management to make many changes (especially ones that seem focused on just reducing your stress level) if management is indeed satisfied enough with how things are.
The idea of just accepting such poor scoping, endlessly without any care, is just accepting tremendous stress because some people aren't doing better at handling the poor scoping. Don't plan to comfortable settle with an environment of routine, expected-to-continue loose/poor scoping.