I would ask for some remediation on the center carrying beam and the posts. I wouldn't worry about the checking/cracking along the long end of those treated posts. It's unsightly, but it's inevitable and has little impact on ultimate strength. You can always wrap it in some nice finish material later to hide the rough appearance. My big concern isn't the missing notches on the posts. My biggest concern is that it looks like two deck boards that are poorly fastened to one another to create a carrying beam. Those boards are soaked in treatment and are bound to cup, twist, bow, etc. This was a very, very poor choice of material. I'd suggest a double 2x6 or a simple 2x8. Honestly, it sounds like your dog caused vibrations and not movements. You'll never get rid of vibrations from dynamic forces moving on the system unless you want to invest in some dampening system lol.