Acknowledgment of funding and adding an affiliation in exchange of permission to use experimental data? Since it is required to be precise and ask one question at a time, I had to split my problem into two. The first part (background) is mentioned here:

Gifting co-authorship if the topic of research was suggested by my boss?

The second part is here. So, I had finished 40% of a paper 1 as a postdoc in institute X. And, I had resigned from X on 31st of May. After completing the paper 1, I am planning to work and write paper 2 which will be an extension; probably a much advanced version of paper 1. I will start working on paper 2 from July/August and probably submit it in November if everything goes according to the plan. But for paper 2, I need experimental data and even though I had many options for sources to get the data, I thought it was better to seek permission from my boss at X before I use their data. 

Also, note that the topic of research for paper 1 was was suggested by my former boss at X, please see the link I mentioned above.

But my boss at X has two conditions: First, I am required to have affiliation of his research unit for paper 2. Second, in the acknowledgement section, I will have to strictly mention that the research of paper 2 was funded by a project grant from his research unit.

In reality, he had the data from the year 2008, from a private seismic survey company XYZ and such data is very regular thing to obtain in any seismic survey. I have such data as well from my previous job profile as a geophysicist. I can easily get more of such data and permission to use them from other companies without any hassle.

As a result of this conflict my relationship with him has also got strained. I tried to clear things with him through a meeting, sent him two emails, but he did not replied. He has probably burnt the bridges and avoids confrontation. He may also be upset from me as I am going away from his clutches. He had even restricted me that I cannot contact the company XYZ directly for asking them for the permission to use the data. 

Is this not a form of bullying? 

Also he made it sound as if I was stealing their data, on the contrary, I was merely seeking permission to use them. I was kind of miffed with that, not acceptable to me.

I was thinking, that if I am allowed to use the data, I will acknowledge the company, my boss at X and his research unit. Also I would acknowledge him for suggesting me the topic of research for paper 1. Regarding the data permission, I have done this before as well in a different institute with another professor. It was never a problem. A simple acknowledgment sentence in the paper has been sufficient for such purposes.

So how fair are the demands of my boss at X?
From what you describe here, the requests for the paper are fair: nothing is asked but a proper description of what you did where, and who funded you and who provided the data that is analyzed. 
The crucial point is that you say paper 2 is an advanced version/extension of paper 1 you plan to do using data you got at affiliation 1, which I read as  both papers are part of wrapping up your scientific output at affiliation 1.

(To me this sounds as if paper 1 is describing the theory and paper 2 describes the application to experimental data - if that's the case, you have two papers closely related to project 1.
Also, how closely related is your project at affiliation 2? The more different that is, the more related to affiliation 1 will paper 2 be seen.)


  I am required to have affiliation of his research unit for paper 2. Second, in the acknowledgement section, I will have to strictly mention that the research of paper 2 was funded by a project grant from his research unit.


At least in my field (not geophysics) it is perfectly normal to list an author who changed affiliations with 2 affiliations. From what you say, you'll anyways need to refer to paper with something like "this is an extension of [paper1]" in the introduction. In addition, you can clarify your affiliations in the acknowledgements saying that you started working on the project while at affiliation 1, funded by project 1, and now are at affiliation 2, funded by project 2. If you like, you coul dbe even more specific in describing what was done where.
If your project at affiliation 2 is unrelated, you may give affiliations as "1, now working at 2".


  In reality, he had the data from the year 2008, from a private seismic survey company XYZ and such data is very regular thing to obtain in any seismic survey.
  I have such data as well from my previous job profile as a geophysicist. I can easily get more of such data and permission to use them from other companies without any hassle.


Whether you could have used other data as well does not matter at all. The point is that you give the proper origin of the data you actually use.
And, regardless whether you use data you got at affiliation 1 or at job 0, you need permission by the respective employer to use that data for new studies or papers. (I'm talking German legislation, things may vary somewhere else, but I guess this is something you can safely extrapolate to other parts of the world).



The only thing you describe that sounds somewhat unusual to me is that prof. 1 asked you to no not contact company XYZ on your own. But we'd require background information by prof. 1 to judge whether this is fine or not. 

I do have the impression from your posts, though, that you are still thoroughly stressed about your situation and still too upset to think calmly. One point that triggers this impression in your post is that you write in length about prof. 1 being upset - and are - judging from what your write -  thoroughly upset yourself, but never mention taking into account even the possibility that you are also upset.
Seeing this, prof. 1 very clearly spelling out the IMHO usual points in wrapping up publication plans after a change of affiliation may even just be a reaction to you being as upset as you are. (Which does not mean that prof. 1 is not upset, but we academia.sx readers of your question do not have the neceassary background information to judge who else but you is upset.)