Depends on the will. If the will is silent there is a presumption which differs in each state.
JT is a disaster and generally not recomended. What about death and bankruptcy - easy to lose the whole house instead of just 50%.
Yes it might be better to pre-empt any messy issues by selling beforehand. I have seen contracts signed and then a caveat lodged afterwards causing settlement to not occur.
but get advice.
infidelity is irrelevant under Australian laws,
Your parents will be in for a bumpy ride. Any transfer now to defeat the entitlemnet of the spouse could lead to issues. But it still maybe worthwhile.
Seek legal advice asap.