In all of these the cock is France, the dog is likely the USA, Australia, and/or the UK.
鸡飞狗跳 ji1 fei1 gou3 tiao4
lit. “the cock flies, the dog jumps” i.e. great commotion.The dog here is a pet dog, not a wild, work, or war dog.
The jump here implies omens.
The Cock flies (away) while the dog jumps omenously.
Likely: the dog is displeased at the flight of the cock.
This one indicates stress between the USA and France.
偷鸡摸狗 tou1 ji1 mo1 gou3
lit. steal chickens, pet dogs i.e. placate guards while seizing that which they protect.
This is likely a reference to the recent attempted theft and sale of nuclear secrets, the idea is that the FBI should be bribed, cajoled, mocked etc. to distract from and calm down the obvious fallout. The chickens in this one is likely not France but instead is “merely” United States’ state secrets.
鸡犬不宁 ji1 quan3 bu4 ning2
the cock and the dog are not peaceful
i.e. the USA and France alike are hostile toward the PRC.
To further decode these look for homophones e.g. 集权不宁 especially those which match by tone.
几 机 犬 are all first tone.
Note also that 鸡 is also slang for a prostitute.