Yeah you'd wanna run most every day, 5 days/week if possible but you might have some things to fix to get there.
That you feel better when running faster could be a clue that your IT band pain is actually psoas weakness. That knee lift is taking some load of the psoas...
I also wonder if your quad and calf injury are related -- connection would be tight psoas entrapping a nerve at ~L2 and when quad compensates (to do the work of the psoas), your femoral nerve is getting squeezed also, though it could just be straight up week psoas. Did you atrain the quad at the start of a faster interval? That'd be a clue here also.
the shin problem could be a limb-length issue related to the hip flexor issues. Couple of ways that might play out.
the compression in the spine would also worsen the whole system. Have you thought about some traction table? Or some other sort of inversion therapy? Water running would have a similar effect decompressing the spine and sciatic nerve.
from what you report of the recent training, you need to get into the pool and do your running there, though I would not be surprised at all to hear you have no troubles on stairs which might actually help your back. Either way you have to correct the strength ratios in the hip flexors. A 2kg medicine ball might be useful to test the hypothesis, as you can massage the psoas by putting your weight on a 2kg ball set to press into the psoas, just next to your spine. (you would be in a prone position for this)
If you email me I can write out a more thorough explanation for those strengthening exercises.