There's a definitely a difference in "good practice" and "code".
Rattler12 said:
Is it wise to put a smaller rated breaker in the line feeding a larger rated breaker? I can see a 75 amp breaker feeding a 60 amp but not visa versa.
Generally, yes, it's good practice to coordinate protection where the closest device trips first. Especially in cases where you're trying to limit the number of affected adjacent systems. A lot of my early career was spent doing coordination studies for power plants - it was a fun puzzle to try to figure out in complicated systems.
But in this case, it's probably worth trying a larger device downstream. We're not trying to coordinate a complex system and the house panel breaker still provides protection. If it works, good for OP. If it doesn't, put the 60A back in the panel and it makes it easier to reset when it trips.
Technically though, the subpanel doesn't need a main breaker at all. It just needs a disconnect. A main breaker is just an easy way to accomplish this.
Rattler12 said:
Is that 6/3 wire pictured, the feeder from the house box to the shop box? If it is, isn't it rated for 55 Amps?
Yes, that's how I would rate it unless I knew the temp ratings were 75F (In my experience, more often ends up at 60F). I think it's good practice to bump up to the next size, and that's what I would do if I were designing it. But per code, you're allowed to round up the cable ampacity to the next standard circuit breaker rating (with caveats), which is 60A. FWIW, I think the cable is made with enough design margin that it's OK.
Just my opinion in both cases, though.