(Non US national here)
Can the governors of states that disagree with the deployment of federal troops deploy some kind of state force to stop these? Does state police loyalty lie with the governor or the nation?
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There are multiple police forces in the US: city, county, state and national, and each is responsible to a relevant executive. Typical "police" are city police, who are responsible to a chief of police, who is appointed at the municipal level. Counties usually have an elected sheriff, and a set of deputies; at the state level, they are usually called state troopers. The typical protocol is that decisions are made at the lowest applicable level, so Seattle police enforce or refuse to enforce laws within Seattle, and King County police enforce or not in remaining unincorporated locations in the county. Individual city officers do not then decide to ignore the chief of police and instead follow orders from the county sheriff or the governor. However, each state grants vast powers to their governor, so it is possible that in the case of a state of emergency, the governor can take command of all law enforcement in the state.
Governors "can't" deploy police to oppose federal law enforcement in a shoot-out, except that they might actually do so on some theory that federal law enforcement officers are violating the law. In general, you are not immune to arrest for illegal acts just because you are a law enforcement officer. The governor of Washington could easily (in the legal sense) declare an emergency and order state troopers to prevent federal officers from effecting arrests. Of course, resolving these disputes in court is another option.