on a 'Round Oak' oil furnace

What kind of burner is it?

If the reset is pushed many times the combustion chamber can become oil soaked making for nasty situation when the burner does light. The chamber can absorb oil like a sponge so you may not see much oil

The technically correct answer is to replace the chamber if soaked but usually you just let it burn out carefully and with a fire extinguisher handy, taking precautions and using common sense. I have herd of spraying the chamber with a fire extinguisher before starting the burner

I think you need to find a buddy who does oil and have him come over and guide you.

You're not going to like this answer, but you don't know what you're doing. You don't have the skills, and tools to properly diagnose and fix you're problem. I don't know if your an hvac tech or the homeowner.
You didn't diagnose and find the problem. You have a stack relay, pushed multiply times, locking out and you haven't figured out why. Now you can't get it to run right and you're going to bleed the pump?
You threw a lit piece of paper in the chamber-you're really lucky it didn't light off.
You haven't mentioned properly checking the motor, vac/pressure check of fuel pump or any electrical diagnosis.

I wouldn't go to a foot doctor for a heart issue just because they are both doctors-different skills, different diagnostic tools, different experiences.
Please get someone qualified out there. If you are an hvac tech, you're doing your customer/friend/family member a huge disservice.

Looks like a Wayne M assembly, but everything @STEVEusaPA said and a bag o chips.
Do you diagnose gas leaks with a flame?
Where is the house? Should i get my family in the basement?

Your set up will have to do with the nozzle firing rate and pattern. There are no good rule of thumb set ups really. You could start with unclejohns set up but keep in mind with every false start or delay you'll be adding oil or partially burned oil, as evidenced by the condition of your insulators. Start with what unclejohn said.

For electrode gap, there are several dimensions you need to be aware of. Just the gap isn't enough. Gap should be 1/8"-5/32". Electrode ends should be at least 5/16" above the centerline of the nozzle, and about 1/16" in front of the nozzle. (Beckett makes a "T"-guage which makes these settings pretty easy). The idea is that the air from the combustion fan blows the spark out in a "bow" shape to contact the oil spray and vaporize the droplets of oil so they can be ignited. If the electrodes are themselves in the spray, they will quickly become carbon-fouled and you will have ignition failure. By the way-no one on here is accusing you of "being some freaking kid fresh out of high school." However, by your own statement in your original post, you never worked on an oil burner before. There is a steep learning curve, and you can get yourself and your customer into hell's own trouble by not knowing the score before you get into something like this. You've got gas experience-just imagine someone working on a gas appliance with no prior experience-shouldn't take much imagination to guess what could happen to him! There's no need to be offended just because one of us pointed this out-no one is trying to offend you, just trying to keep you and your customer healthy and happy.