Is that true for an open-source algorithm? I'm sure that the Russian Government still uses things like SHA256, even though it's an approved NIST standard.

I would imagine so, particularly in cryptography, being able to view the source is not the same as being able to (easily) verify it doesn't contain any backdoors. See ECDSA as good example; we can see the entire standard and yet we can't be sure that it hasn't been weakened/backdoored by its creators despite it looking secure on the surface. See the start of [0], the bit about the standard pushing state leaking through secret dependent operations.

[0] https://blog.cr.yp.to/20140323-ecdsa.html

Edit: my point here is that America would have to trust that there was no way that the Russians had added backdoors. Given their history and the current political climate I can't see that trust coming soon.

You can go a long way towards showing a standard hasn't been backdoored by showing the parameters were generated by 'nothing up your sleeve' numbers. E.g. start trying RSA keys at SHA-256 of 1. And keep incrementing until requirements are met.

When it comes to military or digital signatures - it's gost block cipher, gost hash functions and gost ec signatures.

No, it does not. GOST algorithms usage is a must inside government structures without any exceptions.

Incorrect, a lot of people don't know but it was the NATO and the united states that was actually actively hostile to russia. (long family history of military folks who had EYES only access.)