Josh:
Thanks for writing in with yor question.
First some general theory:
Sprint training whether hill sprints or flat will develop power. As you probably know power=work/time. Work=force x distance. When it comes to muscles, Force = Strength. If you follow this logic you can see that Power is a function of Strength. When it comes to running faster it is far less importnat to be strong than it is to be powerful. That’s why the strongest athletes like power lifters are not the fastest. Strength training may improve your power if you are especially weak. If you can’t DL your body weight then you can probably become more poweful just be increasing strength. At some point in the development of strength the gains to power diminish and you will need to shift to power training. When this occurs is anyones guess. Many thousandfs of words have been written about the benefits of general strength training to improving sport performance. I’ll save you some trouble by telling you that no one really knows the answer. For more advanced runners I work with I lean much more on power training for performance improvement and use general strength for durability and injury prevention.
Now to answer your question:
You can do the max strength and sprints with tire on the same day. Do the power work first and then the max. Normally I would put these two workouts on different days separated by at least 72 hours. Normal periodization would be for 8 weeks of max strength then 8 weeks of power/speed. In your case with ultras max or pure speed is less important. The gains from the sprint training come in the form of running economy (increased stride length) at lower speeds. Add up a 2cm average increased stride length over several hundred thousdands os trides taken in an ultra and you can see where this pays off big time.
I hope this helps.
Scott