I didn't explain the multi-kill portion of the question... though they are unrelated to the yellow circle inserts. You can get your multi-kill weapons a variety of ways, a couple of the easier ways are to complete Bicycle Girl and collect to Well Walker Jackpots. I'd consider those the low hanging fruit variety... beyond that completing each mode awards another and I believe some reward two, letting the mode time out will not award a MK weapon, you have to complete the mode, or make a certain amount of progress, like on tunnel, you don't need to make all the shots, I think 3 shots awards it, but you just can't craddle the ball and let the mode time out. A couple of the others are completing two sets of drop targets while in Blood Bath and collecting a specific number of supplies... I think the default is 9... so complete the drop targets 9 times and that will award another multi-kill weapon. Get all your multi-kill weapons collected and shoot Woodbury to start Horde.
Edit: As far as Seige goes... can't help you... never made it myself before I sold the game