What approach to gears and so on gives the maximum speed for the least sweat?
Most cyclists sweat because they are generating power to progress and will sweat more the warmer/humid it is. They also generate 'body' heat as a by-product of the power generated for progress, to an individual degree. The less clothing the rider wears the more the cooling effect of the apparent wind (so definitely leave that helmet at home
@mjr ). The faster a rider cycles the more the cooling effect, but conversely the greater the body heat created by the power needed to counter air drag (the latter increasing to the power of 3 with speed). The air cooling effect increases with a lower exponential than air drag. So the answer is, as
@Drago has said, 0 mph. To achieve this any gear may be used. More practically, go as slowly as your busy appointment schedule allows. Wearing heart rate monitor and riding slowly enough to keep it below an individually set figure would set an arbitary but specific limit. You'd also be able to experiment with whether lower cadence and a higher gear keeps your heart rate down better than spinning in a lower gear. Going up a decent hill at a specific speed I know that my heart rate will stay lower if I spin at higher cadence rather than push up at a slower cadence with more force on each pedal stroke. YMMV