I have the same shocks on my 99 Accord...and I love them, they are great.
The thicker rod of the Koni's don't allow the use of previous hardware like the washers. That's why the Koni's come with their own hardware to replace the stock hardware with the smaller hole in it.
The "yellow-orange squishy thing" is the bumpstop. That will still work with the koni's...there is enough play in the bumpstop to push it over the Koni rod.
Off the top of my head the Koni's have their own washers like you said, but also their own metal sleeves that go in the stock top mount rubber bushings.
You could probably drill out the washer holes to work with the Koni's, but you couldn't do that with the metal sleeves that go into the stock bushings (again...like the bumpstops, you just push the thicker sleeves in there, it's tight, but it fits).
You might want to see if you can get the hardware separately from Koni. If that's not possible, or you want to do it yourself, I bet you can find some hardware on your own that would work. Might have to cut some piping for the sleeves, etc. (Only do this if you feel confident you can make it work haha...I don't want you to mess up your car).
For the clips that go into the groove...you don't need ring pliers...they woudn't work...there is no hole in the clips. You just need two pairs of pliers (or subsititute one with vise-grips, etc). Just pull them apart slightly and slide up or down.
Hope that helps and if you have any other q's let me know.