There are two parts to your question
How to deal with stress
This is largely out scope for this forum. However, you can find various answers on other stack exchange sites including
there is always that expectation to deliver on time and at high quality leading to immense pressure and angry clients/stakeholders if it's not met
This is the crux of your issue - you are asking for techniques to alleviate stakeholder concerns and also to generate flexibility in planning.
I am going to purposefully refrain from taking this into an Agile versus Waterfall direction which is where it will inevitably end up.
For now; let's break down your points logically.
As the Project Manager you have a responsibility to communicate a realistic and sustainable delivery mechanism for the project. If you are not well versed in the Iron Triangle then start here as your baseline.
Essentially it can be summarized that if at anytime either scope or time or cost is fixed then the other parts of the triangle must flex.
Agile achieves this through flexing on scope and waterfall project management traditionally flexes on time at the end of the project through extension and inevitably flexing on cost.
To cope with this you have several tools at your disposal
Your key message should be that you are communicating a range of estimates and ensuring that stakeholders fully understand the results of their decision making.
Most managers and sponsors can cope with bad news if they are presented with range of responses.
In addition, your key ally in dealing with stress as a project manager is your
It is almost impossible to become stressed about a project if you have correctly identified the risks surrounding the delivery and communicated those effectively.
I would also ensure that your communications structure is fully understood and that all meetings are effective and relevant.
I have designed the following images for you as a prompt to put that rigour around your project.
Communicating Effectively as a Project Manager
Effective meetings for a Project Manager
My last piece of advice is simply;
You are not alone as a PM. It may feel that way but you have a whole community of people in exactly the same lonely boat doing the same job under the same constraints. Reach out - you can blog, write, make SlideShares, capture lessons learned and work with peers. Don't keep your stress to yourself.
One of the key reasons for being a PM is to exert a level of control of your career and that means achieving a healthy work-life balance. No matter how stressed a project gets I never forget that my kids need taken to the cinema at the weekend and I am having dinner with my wife.
It's just a project, simply a big to-do list.