Does this circuit have a counting sequence of : 000, 001, 011, 111 ?

 

If the 3rd output is from Q (instead of !Q used for input back to 1st FF), It will not count as you anticipate.

"Correct" binary counting is 000, 001, 010, 011, 100, 101, 110, 111 <rollback to zero> This is counting from 0 to 7 with 3 bits.

There is a chapter in the e-book here that covers this exact problem.

Do you have Logisim or other logic simulation software? It is free and can come in handy for these type of problems.