I've always used WD-40 and/or Brakleen for cleaning, but lately I ran out of WD and had a can of Liquid Wrench w/Teflon sitting around, so I used that. You shoulda seen the black crap that came out of all the nooks and crannies! Plus, it's got Teflon

I also use generic automotive grease, I got a tube of moly bearing grease from Lordco for $10 and it'll probably last me the rest of my life

Never thought about a syringe though, I always use my fingers and make a HUGE mess, I might have to pick up a few syringes this weekend
ATF is an excellent choice for gun oil, I also use it on my paintball guns since it's designed not to eat o-rings while also effectively lubricating and protecting aluminum from wear (not an easy task, aluminum can be wierd stuff sometimes). There's a lot of aluminum-to-steel contact/movement in an automatic trans as well, which frequently occurs at high temperatures and speeds, so if it's good enough for a Torqueflite it's good enough for my SKS

15W-40 Diesel engine oil or 20W-50 will also work, as long as it's clean. I've been meaning to try some moly engine assembly lube in my rifles, and moly gear oil. It's usually pricey, but anything with molybdenum disulfide in it is awesome for wear protection.