AS far as 3 engine planes go, there were 2 reasons, which overlap a bit. Also props come into play.
Twins are an obvious way to get more power.
However a twin with an engine out has problems, especially with primitive propellers (fixed pitch, two pitch) and only somewhat better with constant speed. You not only have asymmetric thrust you have either a prop windmilling trying to turn a dead engine or several sq ft of airbrake sticking into the air if the engine is not turning (really broken engine). Many or most early twins could not maintain altitude with one engine dead.
Thus the popularity of 3 engine planes. with 2/3rds the power, even with crappy propellers, the plane could stay in the air and not just extend the crash distance. Note that this only applies to a plane in cruising flight. Loosing an engine while taking off is still in a world of it's own. Ceiling with an engine was also important. More later.
Engines and airframes tended to zig-zag back and forth a bit. Sometimes the engine makers were ahead and sometimes they were behind. Sometimes the airframe makers wanted more power but did not want a single 1000hp engine (1920s) as the vibration and propeller requirements (size) were more than what the airframe makers wanted to deal with.
Sometimes it was the cost/availalabitly of the engines.
The Stinson company built 31 of these starting in 1934
with 3 260hp Lycoming engines. This was cheaper than using 2 400-450 hp P & W engines or Wrights. Note the ground adjustable propellers.
Some went to Australia where the 3 engine "safety" was a selling point. The two survivors were repowered with 450hp P&W's during WW II when they could not parts for the Lycomings. The change to fulling feathering propellers was dramatic. The Lockheed 10 with a pair of 450hp engines was good for 4,000ft on one engine, Using fully feathering props it could stay in the air at 10,000ft at the same weight. Airlines (and warplanes) had to take into account the terrain and weather conditions they were flying in.
With lower drag aircraft
same 8 passengers, 2 crew as the Stinson, cruising power could be much lower than take-off power and the better props allowed for matching engine rpm to speed much better.
There was a larger "reserve" of power to be used if one engine went out.
There was a version of the Lockheed 10 pictured above that used 550-600hp engines for airlines that flew in mountainous areas before full feathering props were common. Got worse gas mileage but they made the trade for safety.
Some tri-motors were a little shaky, some in the 1920s and one in the late 40s/early 50s
Somebody thought that this was a good idea, take the two 6 cylinder engines off a DH Dove and replace them with three 4 cylinder engines and shorten the fuselage a bit.
Plane shown has fixed pitch props to get around the problems they had with the variable pitch props. However the change just made the under powered condition worse.
7 out of the 20 aircraft built were repowered with Lycoming flat 4s of greater power and with full feathering constant speed props.
In the US in the 20s there was craze for using different sized engines, one large on the fuselage and one smaller one under each wing. That that did for handling with an engine out can only be guessed at. All 3 engines turned the same way.