Let me be blunt - your job isn't to keep learning, your job is to perform.
Yes in IT we do need to learn new stuff to keep up, but that has nothing to with the actual tasks you are asked to perform. Companies don't pay for people just to learn and they actually don't care if you learn anything if you do the work you are assigned well.
You went to a new team and haven't earned their trust through performance yet. Why on earth would they want to assign you more work if you have made mistakes in what you were given or if you are continually asking for work that you don't already know how to do? Assigning work to you is a risk and it is apparent from their behavior that they don't believe the risk is worth it.
Now clearly some of this is because they do not appear to be the types of people who see value in mentoring and part of this is due to your own poor performance and your inability to play office politics effectively. Truth is though, when you are junior or new, the onus is on you to adapt to them and not vice versa. If this isn't how you would run a team, then fine, don't run a team that way when you're the PM. However, you are not and you need to adapt to how they do business. And you need to get out there and do some serious fence mending before you get fired.
First action is to be meticulous in the work you do for them from now on. No serious mistakes at all and no minor mistakes if possible. You have the time to do things right the first time. So do it. No excuses. You have to impress them and you haven't done that so far. There is more latitude for mistakes once you are trusted. There is none in your current situation. This includes doing things the way they want them and not the way you would prefer if there is a professional disagreement. This is not say you can't disagree professionally ever, but you can't until you gain their trust.
If you are as you say the only person with the right background to be assigned to the project, then there must be tasks that you are qualified to do that they are not. Ask the PM to be assigned to those tasks. They brought you there because of a particular skill set, so ask for tasks relating to that. DO NOT UNDER ANY CIRCUMSTANCES mention wanting to learn new stuff or needing to learn new stuff to do a task. Concentrate on helping with the skill set you currently have until you are trusted.
Talk to them about how and when they want you to communicate with them If the senior turned Lync to Do not Disturb, that is a sign that you were being a pest. This is not the way to earn trust. You need to have sensitivity to their schedule and needs as well as your own.
Do ask for work every day and ask the PM not the senior dev. If he tells you to ask the senior dev, then ask him and when he doesn't give you any work, tell the PM that you are not getting anything and to please assign something. Volunteer to do the tasks no one else wants to do. Yes this is scut work, but it has to be done and if you do it to an acceptable level of performance you will build trust and you will be building a basis of having done them favors that entitles you at a later date to ask for favors in return.
You may very well need to clear the air. Right now, you know they don't appear to like you or want you on the team. What you don't know is exactly why that is or what you need to fix to be a better fit for this team. Sometimes, it is not that they dislike you but that they are just too swamped to handle the extra task of bringing someone up to speed. Or perhaps you have made some mistakes that made them lose trust or maybe they just didn't want another person on this team. Or maybe they don't actually know what skills you have that can help them out. They may not even know that their actions are bothering you or that you aren't delighted to have nothing to do. They may hate your personality or think you are too pushy. Who knows what they think.
But you can't fix a problem until you know what it is. So I suggest you tell them that you are concerned about how you are doing since you don't seem to be progressing to more important tasks and then ask for their feedback on your performance and what you need to change to make it possible to contribute more fully to the project.
What you hear back may be painful. If they were happy with you, there wouldn't be this issue. Some of what they say may be true and some may not and some may be a matter of having different perspectives. What ever they have to say, hear it out without any defensiveness or attempt to respond until you have heard everything and had a chance to process.
You can't fix what they don't know about, but even if you disagree with their perceptions, they aren't going to change until you know what they think and then make a plan for fixing the perception, if not the actual performance. You have to be aware of perceptions as much as actual performance because they are going to judge your performance based on their perceptions.
In judging performance, perception is all there is. This is why people who quietly work in the background and never have an issue fail to get the rewards they deserve because no one above them perceives the work they did as important or difficult. It is why some people get rewards when they don't appear to many others to have contributed much. They have taken the time to manage the perceptions of those above them though and so their bosses think they are great.
In the long run, while you have to try to fit into the team the way they run it, not everyone is a good fit for every team. You may or may not succeed in turning this one around. At some point, you need to look out for yourself and move on if the team isn't working for you after you have tried to really make it work. I am not a fan of the "run away at the first sign of problem" mindset that some people have, but you also don't want to stay in an environment where you are not happy and they are not happy with you and you have genuinely tried to fix that. In this case, it is not your problem that they would lose the client if you left, so don't feel guilty. If that is actually true, then it is up to them to make sure you are happy.
