Another option is to connect with an already-established 501(c)(3) organization. You may be able to fit under the organization's "umbrella," meaning you can collect funds for your Russian organization through the already established 501(c)(3). That way you wouldn't have to go through the complexities of forming an entirely new 501(c)(3) organization, but donations to your cause would be deductible. Your organization would receive the tax-exempt status of the 501(c)(3) organization.
However, there are certain administrative hassles involved with this type of arrangement. If you're interested in exploring this route, you should seek out already-established 501(c)(3) organizations that may be willing to accept your organization under their umbrella. You'll have to work closely with the 501(c)(3), so you should take your time to find one that fits with your objectives for your organization.
You should also consult with an attorney since any tax-related transactions can be messy and, often, very complex.
A 501(c)(3) is the usual means by which deductions can be taken for contributions and the organization can avoid paying income tax, along with other benefits. You can certainly raise money without this classification but the contributions will not be deductible and there could be income tax consequences. You could create a nonprofit corporation in Washington and not seek 501(c)(3) designation. You could keep the organization here regardless of your moves or you might register it to do business in another state. I do not see how your moves from state to state would affect you qualification as tax exempt. It sounds to me like you ought to re-evaluate how much your moving would really affect the desirability of obtaining tax exempt status. There a so many considerations here that a quick pat answer is not terribly helpful.
This business of sending money overseas is dooable, but dangerous. Since 9/11 these things are scrutinized. FBar and 6038D are on scene.
You can raise and collect money and forward it but your donors won't get a deduction because you won't have a 170c letter. You will have to report the transaction to treasury currency control as you are aggregating dollars.
My best advice is to simply donate to American Dogs and let the Russians donate to Russian Dogs. Nonprofits aren't 100% efficient, a lot have heavy staff, but all of the money you spend setting one up is money that could have been employed in your desire to help actual dogs.
The is amplified by your moving often. We in the California Bar have had a "reorganization" proposal on the table to enable non-profits to move and reorganize as easily as ordinary for profit corporations, but it has not happened. Each state move would means starting all over.
Good Luck with saving dogs!
Curt Harrington Patent & Tax Law Attorney Certified Tax Specialist by the California Board of Legal Specialization PATENTAX.COM This communication is general information and not legal advice, and does not create an attorney-client relationship. This communication should not be relied upon as any type of legal advice. Please note that no attorney-client relationship exists between the sender and the recipient of this message in the absence of either (1) a signed fee contract and (2) remission of an agreed-upon retainer. Absent such an agreement and retainer, I am not engaged by you as an attorney, nor is any other member of my law firm.