Step 1: The task.
We check four statements about muscle proteins and keep only the correct ones.
Step 2: Statement I, actin.
Actin makes up the thin filament, and the thin filament lies in the I-band of the sarcomere. This is true.
Step 3: Statement II, myosin.
Myosin does form the thick filament, but its ATPase (the part that splits ATP) is in the head, not the tail. The head also forms cross-bridges with actin. So this statement is wrong.
Step 4: Statement III, troponin.
Troponin binds calcium ions. When it does, it shifts tropomyosin away and uncovers the active sites on actin so the myosin heads can attach. This is true.
Step 5: Statement IV, tropomyosin.
Tropomyosin only blocks or unblocks the binding sites. It does not break down ATP. ATP is hydrolysed by myosin. So this statement is wrong.
Step 6: Conclusion.
Only I and III are correct, so the answer is "I and III". \[ \boxed{\text{I and III}} \]