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I've no idea about the wood stains no company will say their products are safe for use on animal accommodation. I know this from contacting them when I had to design a cage for a college assignment. Same with the paints as I researched that for touching up an old cage

Of course people do use that sort of thing on their cages/aviaries but no company will stand by their product saying its safe. I decided it wasn't worth the risk but if you go for it do tons and tons of research before selecting one to use. On that note I doubt the species your planning on keeping will do too much chewing on the wood apart from maybe the ringnecks but you can always solve that problem by blocking access to the wood somehow and giving them more interesting stuff to do and chew.