Have you done a copyright search? If the painting was not registered when it was created, it was still automatically covered by the copyright law and the duration of its copyright would be life of the painter + 50 or 75. If the painting was registered, its copyright had a 1st term of 28 years from the date it was secured and could have been renewed, for a total term of protection of 75 years, which would mean its copyright has expired.
If the work is still covered by the copyright of the artist, as inherited by the artist's heirs or someone the artist conveyed the rights to, it wouldn't affect the the rights of what you want to do, as long as you legally own the painting itself.
Whether or not it's under copyright, you can take your own photo (or hire someone to) and you'd own the copyright to that photo and could reproduce it and restrict others from reproducing it. And anyone else is free to make their own photographic copy of it and restrict anyone else from reproducing THEIR versions.
Whether or not the artist's heirs or devisees own the copyright, you can exhibit the painting or loan it out to a museum if you're the legal owner.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
As usual, Pamela Koslyn has done a great job answering. As you can tell from her answer, you are really going beyond the strict confines of copyright law into the realm of "art law," and considering what you seem to be contemplating, you need to sit down with a qualified lawyer who is experienced in that field and to get the advice you need specifically tailored to you.
You first need to determine if the painting is protected under copyright law. To do that you must know at least two things: was the painting ever “published” and was the painting’s copyright ever registered?
A painting is published when it’s displayed to the public without restricting the public from copying it (by drawing or photography, for example) or when the painting is displayed for the purpose of selling it or copies of it.
A painting sitting in an attic or hanging in a home is not published and neither is a painting hanging in a museum that enforces a “no copying” policy. A painting that’s hanging, or has ever hanged, in a gallery or other public place (where more than “a normal circle of a family and its social acquaintances” gather) is published as is a painting that has never been displayed but has been copied or photographed and those reproductions have been distributed to the public.
If published before 1923, no copyright attaches to the painting. If published after 1923, then an involved set of rules apply to determine its copyright status according to the particular facts of its publication. See: http://bit.ly/8K5GfI
If never published, then copyright protects the painting for the life of the author plus 70 years. If the author is unknown, anonymous, or the painting was specially commissioned, or if the author’s date of death is unknown, then copyright protects the painting for 120 years from the date it was created (which must be estimated if unknown).
If copyright attaches to the painting then no one (including "chop/repro shops") can sell copies w/o your permission. Photographs and drawings of the painting can still be created, however, if permitted contractually, and those reproductions can then be distributed to the public if such distribution is “fair” – which is usually for news reporting, educational purposes, or commentary.
If copyright does not attach to the painting then it’s in the public domain – which means, under copyright law at least, that anyone who gains access to it or a copy of it can reproduce it, modify it, adapt it for other uses, make derivative works from it, or anything else that can be done to it or with it, and then the painting or any work based on it may be distributed to the public.
As owner of the physical copy of the painting, however, you can contractually limit what other people do with the painting. Like any other contract, to make the contract enforceable, you must give the other side something and they must give you something.
As for the reproduction shop, you certainly should enter into a written contract that prohibits the shop (and all employees, agents, etc.) from retaining any physical or digital copies and, in the same contract, include a provision that assigns to you all of the reproduction shop’s copyrights in the reproduction that it creates.
As for the museum, it may or may not permit its patrons to take photographs of its works. You’ll have to decide if you can live with whatever downsides there are to having amateur photographs of the painting floating around in cyberspace or real space. In any event, you should have an intellectual property attorney review the agreement between you and the museum (certain arrangements provide tax benefits, for example, and other considerations are involved as well).
I wrote: If copyright attaches to the painting then no one (including "chop/repro shops") can sell copies w/o your permission.
Additional information: On further reflection, I need to amend that response. If copyright does protect the painting then no one can sell copies of it w/o permission FROM THE COPYRIGHT OWNER -- which may not be you but rather the heirs of the painter. When a painting is purchased (or otherwise transferred) what is transferred is only the physical copy of the painting, not the copyright. If the painting is under copyright then even you may not reproduce it w/o permission. Though your question does not suggest that you intend reproduction, this clarification is important.