Acknowledgment of funding and adding an affiliation in exchange of permission to use experimental data?
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.)