I have heard that it is anti semetic to "deny israel has a right to exist"
Which makes me curious. Is it a generally accepted premise either in law or just by people that countries have "rights"?
I think of rights as something people have.
If countries do have rights, is exisiting one of them?
This is a really interesting question. There's international laws signed that say what its signatory states must do for their citizens and what they can do to each other, which I think at implies that states at least have some sort of rights, but this is undermined by what counts as an extant state. If a state does not have sufficient recognition by other states it can't exercise its rights, which is why we see the unequal application of Israel and Palestines rights to exist and defend itself (if those rights meaningfully exist at all).
In other words, a state exists if it has recognition by the hegemony, and if it is it's granted some minimum of rights. A state does not have a right to recognition and therefore doesn't have a right to exist. This seems like a limitation of liberal geopolitics because even if a state has popular support by its own masses and has the monopoly over violence commonly required, there's no guarantee it will be recognized as such.
What other rights do countries claim to have? And do they articulate with that language?
Taxation, passing laws, incarcerate people, control trade, etc?
All things countries do.
Is there a declararion on the rights of governments.
Generally they have what I understand as "negative rights." It's not that countries have explicit rights, but have implicit rights to not have things done to them through other countries signing treaties saying the won't do things. So theoretically countries should have a right to not be war crimed by countries that have signed the Geneva Conventions, e.g. why many less radical pro-Palestine commentators are literally only asking for the fair application of international law to Israel