Yes, Temple aura stacks with Battlefield Medicine, assuming the soldiers are idle. 

@guerringuerrin also: “if I don’t spend resources, I have more to spend later.”

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Hmm, meaning everything else stacks? For example, Battlefield Medicine and temple aura?

Depending on your strategy, you have two ways to heal your troops.  Active would be to focus on healers and research their upgrades.  Passive would be to focus on garrisoning and Temple auras. Always focus resources only on one way in a match. Most good players focus on the passive way, as healers are far too fragile.

Yes. What does not stack is when you have the healing aura of 2 temples very near, that aura only adds once.