The Underground and Third Ring tickets include the Colesseum -- no
need to purchase both.
Sorry but that's not the case. The coopculture website states that general entry tickets must be purchased separately from the tours. If you have a tour booked, you can just buy them as the reservation desk when you show up for it (bring your confirmation printout with you). They may also be purchased online.
http://etickets.coopculture.it/index.php/en/component/snapp/?view=event&id=28410C15-602F-4539-A210-CE0B6183CEFB
"IN ORDER TO VISIT THE UNDERGROUND IT'S ALSO NECESSARY TO BUY THE COLOSSEUM ENTRANCE TICKET AT THE RESERVATION DESK"
But it looks like the Underground tickets are sold out for your dates
already...
Also not the case: the underground/third-ring tours for July won't go on sale until 9:00 AM Rome time on the third Monday in June if they follow the sales pattern of the past 6 months (not formally announced yet).
Do I purchase a separate ticket for admission & underground & 3rd ring
tour or can I purchase 1 ticket that includes both?
There isn't a ticket which includes both: you must buy the tour and general entry tickets separately. Make the tour reservations for all 5 of you according to age: only your 11 year-old will be eligible for the "free" 2-euro tour reservation: the older kids will pay same as an adult ticket. You will not need to purchase separate general entry tickets for any of them, however. Those will be free, and can be picked up at the reservation desk where you'll pay for the adult general-entry tickets. Bring their passports for proof of age!
There is a very concise and helpful explanation for for purchasing underground tours on this Trip Advisor page contributed by a frequent RS forum poster (thank you, Donna!):
https://www.tripadvisor.com/ShowTopic-g187791-i22-k8145551-o1100-How_to_book_the_colosseum_underground_tour-Rome_Lazio.html