Last I checked, which was a long time ago because I'm lazy and been paying the "mow, blow and go" guys to come once a week, my lawn mower starts up on the first try. Just plug the extension cord in and squeeze the switch on the handle.
I think the newer ones have large batteries so you don't have to worry about mowing the extension cord.
As for the old Plymouth, I use oil with far more ZDDP in it than the car saw brand new. I get it at the local auto supply. Their cheapest store brand multi-weight. Why spend more?
With respect to gas, I did rebuild the fuel system with modern rubber components. Modern for 10 to 15 years ago that is. So far, knock on wood, no noticeable deterioration of the components yet. Just a little tendency to vapor lock after driving at freeway speeds in 95+F for long periods of time. And I did not drive the car too much last winter, so it went over 7 months on one tank of gas with no drivability issues even without using Stabil.
I think a lot of the hype about needing gas or oil additives is just an attempt to separate a bunch of people from their money.
One item on the list of things people worry about on old cars is the use of radial tires on older wheels. Will split the rim some people say. This was an interesting read on that topic: http://www.dbtires.com/rimstress.html Of course, they are in the business of selling radials so they may be a little biased.