Describe main criteria of choosing energy-conserving bicycle that allows getting easily up-hill
Uphill - gears and weight. Extending the lowest gear range till its faster and easier to walk and push the bike is cheap and easy and would be the first thing on my list to do.  

Cutting down weight (of the bike) is an has an exponential cost/gram relationship - you quickly get from $1/gram to $100/gram. Cutting your weight does not require the cash, and in most people cases a very effective way to gain speed uphill. (Shaving 5kg off a bike will cost $1000's, even if possible, many people can afford to loose twice that). 

Although you said no engine, in your situation (hot, uphill, no shower at end, health concerns) a pedal assist electric bike would be a very effective way to achieve your apparent aim. Would you be prepared to rethink that position and consider an eBike?