Can I use TC-ER Cu Awg 4-3 THHN for a shed sub-panel?
Type TC cable is for tray cables it requires a cable tray or raceway.
Smurf tube would be the perfect option inside the home as it is cheap and flexible. Non metallic flex conduit is commonly called smurf tube because it is often blue, but could be other colors orange is usually reserved for fiber but that doesn't stop a user for using it in there home for this.
The cable is wet rated and can be direct buried if rated for direct burial if not it will require conduit for the conditions.
The information can be found in article 336 of the NEC it’s only ~2 pages long so not hard to read up on this type of wire.
When sizing the conduit a single wire or cable is limited to a 53% max fill I did not see tc listed for area so you may need to measure yours and make sure the conduit you use inside the home is large enough. For the cable you are using.