My understanding is that steeping and mashing are interchangeable for crystal/caramalts - the sugars have already been converted by kilning the malts while wet, resulting in a "mash" as the crystal malts are being prepared. They are then dried, so the converted sugars should just flow into your wort solution as you steep. Your dark malts have little convertable sugars, so those a great for steeping. No idea about the oats, though - someone else should chime in as I've never used them.
I think you could easily make up the difference in 2-row with more extract, but I don't know about your oatmeal. I've always done partial mashes with more unique base malts like Maris Otter, Vienna, etc., but I'm not sure about the flavor difference between DME and 2-Row. I want to say I've heard most DME is a mix of 2- and 6- row, but someone else might be able to help more.
Mashing in a 3-gallon drink cooler sounds right, I think that should fit your 5 pounds of grain. However, how are you lautering/draining? If you're mashing in a grain bag and the cooler has a drain plug, that should be sufficient for a partial mash; If not, you can sparge in the aformentioned grain bag with a collander/strainer above the boil pot.
I think in partial mashing a super-efficient sparge isn't your #1 priority, as the DME should make up most of your gravity. When I do partial mash I aim for a normal mash, maybe a but more water if I can swing the room (say, 1.25+ qts per lb of grain, or 1.5+ gallons mash water in your case) and a gallon or so to sparge. I also leave the grain in the strainer/collander after sparging over a pan and throw the last of the wort that drips out into the boil once I get it going (probably adds nothing, but hey. It's still well-filtered wort.).
Since you mention your weaker stove, I assume you're doing a smaller partial boil. I might add only 1/4 to 1/3 of your extract at the beginning of the boil to make sure the wort is thin enough for A) full hop utilization, and B) to prevent extra darkening of your wort through scorching/caramelization, then add the rest of the extract with about 5-15 minutes left just to sterilize it. However, Moose Drool is pretty dark, so color shouldn't be a problem.
For my top-up water, I pre-boil about 3 gallons of top-up water the night before, cool it, then pour into sanitized plastic jugs in the fridge so my top-up water also cools my wort in the bucket. This means I don't have to cool my hot wort all the way down to 65 deg in an ice bath. 2 gallons of 90 degree water and 3 gallons of 40 degree water bring you to a 68.5 degree final temp in a closed system, and since those last 30 degrees or so are the longest to cool, this cuts down on my cooling time by a LOT.