Rajsekhar has a good suggestion regarding the right attitude when answering the phone. Mentally do something to take the "edge" off of the call. Try the old public speaking trick of imagining the audience sitting there nude or in their underwear. YOU have control of at least 50% of the call, maybe more.
Some modern technology may also help. Try getting "Caller-ID" on your phones, even if you have to pay for it yourself. That way you gain another important advantage by recognizing the caller before having to interact with them. You can mentally prepare yourself better, maybe even skip the call if you anticipate a problem or situation you can't or don't want to handle at the time. You can always call back right away, or even later when you are better prepared.
When my stutter was its worse, I had a "full-body" stammer. My head would throttle back, eyes roll up, body shake convulsively, and gag as if swallowing my tongue. I'm surprised nobody tried to give me the heimlich maneuver. It was all I could do to hang onto the phone, much less speak into it.
The bottom line to all the problems you have related is the FEAR of the reaction your listeners are having to your stutter. While this is bad enough when in the presence of the audience, for many (including me) it was worse on the phone where I could not see the listener. I would imagine all manner of ridicule, mockery and other ghastly things going on at the other end of the line.
Most interaction in the workplace and over the phone, is interrogative. Someone is asking a question, seeking information or validating some they already have.
KNOWLEDGE IS POWER!! (remember, you saw it here. ;-) I made sure I knew more about my area than anyone else in the company. Then, I made sure other people thought so. I got on every committee, memo distribution list and discussion group I could find. Yeah, I stuttered, but people NEEDED to know what I knew. I found that I was usually the only one that knew what was important to other groups and I knew who to go to for more information. That is POWER.
I believe attitude has a lot to do with success. I'm not saying be belligerent or talk down to people, but be helpful, become the resource they can trust, and feel comfortable with. Everyone has their own performance issues on the job, some hide it better than others, but every individual is important to the whole, that's why they are called companies.
YOU are important to that caller, customer or manager, and you will find value in knowing them, even if only briefly. My customers (and in the workplace, co-workers are customers) pay me for information. When they call, I become their "friend in the business". I talk to them like a colleague trying to solve a mutual problem. They have information I need and I have what they need. We both have to share what we know to reach a mutually successful result.
That's true in virtually all conversations. Complaints are just questions that need answers and understanding. Most of my clients are doctors. When they call me, I become the doctor and they are my patient. I start by asking lots of questions and let them talk and I take profuse and detailed notes.
Get a small spiral bound notebook, write EVERYTHING in it and carry it with you. Refer to it when you speak. You will be more comfortable because you will be more accurate, and it gives you a reason to relieve the pressure of direct frontal scrutiny by checking your notes as you speak.
When I stutter, they don't mind as much because they have already had their chance to get what they have to say over with. It also gives them a chance to reflect on how little they know (or even care to know) about the situation and confirms the value of my company and services to their success.
I look for something in their story that can be humorous and try to get them to see the comedy in the situation, if possible. I NEVER make any disparaging comments about ANYONE, and NEVER repeat things that they don't need to know, or I have not been authorized to pass on. Basic good business sense.
Most of the time, once I have their input and know what they want to get done, I can thank them for calling and tell them I'll get back to them later or connect to their computer. They are always grateful to be able to get on with their own priorities, and back to what THEY do best, healing people.
I have a client who stutters so much that no one on my staff could understand him, nor had the patience to listen very long. They forwarded his calls to me. The first time I talked to him, I explained that I also stuttered. It was like a new person was on the phone. He took a deep breath, paused and instantly became fluent and relaxed. Now, he likes to call so much he always asks for me and is no longer afraid of the "unknown" judgmental audience at the other end of the line.