Sorry for the slow response, but when in the process of taking out the breakers to investigate the bus bar I figured out what was going on and it took me a few days to confirm it. The two bad breakers were in the same row of my sub panel, and it dawned on me that if one the lines from the main panel that feeds sub panel was bad, that could be causing my problem. So I tested the two breakers (tandem) in the main panel that were feeding my sub and sure enough one of them was putting out 25V. So I replaced the tandem breaker in the main panel and now everything is working agin.
Thank for your suggestion about the breakers being in the same row, which got me thinking about the problem in a different way.
Number 4 is the big clue. One side of your electric service is dropping out, turning on the stove causes it to back feed. This is likely caused by a loose connection on one of the service lines feeding your house. There are however a few other possibilities I'll cover below.
For the sake of explanation, let's say the location of the problem is a loose lug inside your electric meter. Over time that loose connection has been building up heat, it's likely aluminum wire that melts and burns and produces black carbon. As this continues over time it eventually works its way "open" and will no longer feed that side of your electric service resulting in partial lights out in your house, that might mysteriously just come back on as that burning lug cools down. You turn on the stove (which uses both sides of the electric service) and it back-feeds current through the circuit back to that bad lug, when that happens and the two opposing sides of the electric service meet it arcs and it can actually create a weak "weld" of sorts, restoring the power again, until the connection once again burns open. repeat. Eventually it will just burn open and be broken until someone finds the problem.
Connection points on the power companies side include the pole (or underground connection vault or stump), and the meter can, and possibly an outdoor disconnect. Most often in our area the problem is one of the meter lugs was not tight.
On the customers side, where the service feeders connect to the main breaker inside your panel is also a place for a possible loose lug. (In my area the power utility will actually tighten main lugs in the customers panel, many utilities however will stop at the meter.) Beyond that it's also possible there is a problem with the main breaker or panel buss itself, but that's less likely (IMO) than all the above.
Start with a call to your power company and report "intermittent partial light out". They will check connections outside your house and may even check your panel for loose feeders. They also may not check past the meter, in that case you would need an electrician to check your panel for loose feed--if the power company didn't solve the problem outside.
Most electric utilities don't charge for the service call for their side, if it is your equipment in the end, you will of course have to pay an electrician for their work.
Best Answer
There may be no damage to the outlets but there will be damage at a common location possibly even prior to the lighting circuit tap.
What is usually the problem with a large load added to other devices is there is a failure most commonly at a back stab (the push in connectors on receptacles and switches) the next most common place is a loose wire nut with a loose wire. A broken wire at one of the locations and least often a problem in the service panel.
Work backwards towards the service panel the problem is at the last working device (switch or receptacle) or the first non working device. The failure will be on the hot since the breaker shows good but no power at the receptacles.
Kitchen countertop circuits are normally 20 amp and the lighting is not normally them. And backstabs are not for 20 amp circuits so it is probably a bad splice or broken wire, checking at the light switch connections and working to the service panel for the bad connection is the task. Most probably the hot.