Goals Realistic?

you mean that you're walking a 17-minute mile, not 3 miles in 17 mins?

people do walk/speedwalk... if you're serious about doing a sprint tri, you ought to get a bike. the stationary bike, while good exercise, is not much like riding a real bike.

I think that early June might be pushing the envelope a little bit. Are there any tris near you in July or even August? The shortest sprint program on this site is 13 weeks.

edit: I just think that waiting and building up a better base might allow you to enjoy your first tri, rather than surviving it.   

You can do it, especially if your goal is to finish.  You have 6+ weeks to add more volume to your walk/run and bike portions.

Do you have a bike you can use?  Can you get out on that at least once a week to ride for about an hour?

Lots of people walk during the "run" portion.  Any forward movement under your own power is legal.   

There are lots of people who run/walk the run portion and even walking is fine.  You'll find that people at Triathlons are generally very supportive and non-judgemental.    Pretty much everyone is happy to see someone finish the race no matter how long it takes.    I would probably suggest taking your run and bike workouts outdoors to get a feel for real distances.  The speeds and difficulty levels on treadmills and stationary bikes can be deceiving.  

Please consult with your doctor (crossing my fingers, hoping you have already...)

If the rest of your introduction is correct (not doubting it) then you can certainly do a triathlon in early June.  I've walked during part of the run on EVERY triathlon I've done (including several miles during my HIM).  At least one regular here on BT speedwalks races and is faster than many "runners." 

When you talk about goals, realize that you are talking about your specific situation, not an average of some anonymous group.  At one extreme, someone could go from couch to IM in 3 months IF that was the most important thing to do during those 3 months (quit work, hire domestic help, big $$ on gear, etc.)  They might not be fast, but they could train enough to be confident of completing it in <17 hours.  On the other end, many of us started out barely able to commit to training for a sprint (jobs, kids, household duties, friends, TV, etc.)  Until you make the goal of completing a sprint triathlon more important than those last few "priorities" even a sprint tri is "unrealistic."