Do I need to add grounding and replace breakers at my sub-panel?

If your 15A & 20A circuits are 120V, you are not fully loaded on a 30A 240V feed IF they are on "opposite legs" - USA/Canada power is 120/240V and a 30A 240V feed can run 60A of 120V circuits fully loaded without tripping. Half the 120V circuits are on one hot "leg" and the other half are on the other hot "leg" and there is 240V between the "hots" of opposite legs, and 120V between the hots and Neutral. When oposite legs are loaded at the same time, neutral carries only the difference between them, not the sum - so it all stays below 30A in the wires.

But the broader concept there is that it's perfectly fine (and normal) to have breakers that add up to more than the feed amperage - of course, you'll trip the feed breaker if you use too many things at the same time, but that's also why it's OK - the main feed breaker will protect the wire. You may well have a 15A breaker on a lighting circuit that never draws more than 5A, (or 1A in a 10x12 shed with LED or florescent), for instance.

You should indeed have not one, but two grounding rods at the shed, 8 feet or more apart. The reason for two is that one grounding rod is acceptable if you do a test that costs more than a second grounding rod and 8 more feet of wire, but with the second rod you don't need the test.

There are some probable code issues with the power feed itself.

10/3 cable with 4 wires is not cordage (which for the same 4 wires would be marked 10/4) - it's wire that's intended to be installed and left in a protected situation, not moving.

If that cable is lying on the ground, it would need to move either underground into a conduit (where, in fact, you would not want cable, you'd want wires - cable is VERY hard to pull into conduit) or lashed onto a suspension cable overhead, out of reach. The cable or wires used must also be "suitable for wet locations."

For a "tiny house" on wheels that might move, you might want a flexible, tough cord that can be left on the ground - "cordage" - but that's really not appropriate for a shed that stays in one spot in the yard, and in most cases even with a "tiny house on wheels" or RV it's preferable from a damage point of view to run conduit out to a pedestal where a short cord can connect to the thing that may move.