How to set the wired Xbox controller to player 2

Lazarus starts with different items than Isaac does. Notably, he starts with an item that gives him an extra life (hence the name Lazarus).

He also seems to start with a random pill... but unlike Isaac, Lazarus doesn't start with any Bombs.

Edit: I haven't gotten far enough to check this, but in the original game, Isaac would start with The Dice if you'd unlocked them... Lazarus likely doesn't.

Yes. Here is a number of ways, in increasing order of ridiculousness:

• Double use items. Some spacebar items will trigger twice in co-op mode. This works for items like Mr. Boom and Yum heart (when co-op baby has a half heart). It does not work with items like Prayer Card or Book of Revelations.
• No-tear challenges. The no-tear challenges would prevent you from destroying poop or putting fires out, but your coop baby can fire tears normally. This is especially useful in a few challenges such as The Bean, where your coop partner really is your only source of firepower, or in Solar System, where your coop partner can fire things from a distance (with its own blue attack fly) while Isaac cowers safely in a corner. This does require you to have more than one hitpoint to spare for your coop baby.
  I literally beat those challenges while using WASD in a hand and the Start button on my controller in the other.
  Caveat: Rainbow poop does not heal the co-op buddy. Bonus: In Cat Got Your Tongue, your co-op tears make your main player generate flies.
• Health management. Have a bunch of blue hearts and a black heart is on the ground? If you pick up the black heart directly, it'll go last on your stack of hearts and you might lose it at an inappropriate time, such as right after having cleared a room (and this happens every. single. time.) What you can do instead is hitting Start on your controller, collecting the black hearts, then hitting start again. When you hit start the first time, some of your blue hearts are transferred to your coop partner. When you hit start again, these blue hearts go on top of your health bar, providing helpful "padding" and increasing the chance that they actually damage a boss.
• Sacrifice rooms. If you have a single black or red heart, you can use the co-op buddy to use your heart containers yourself.
• Loot surrounded by spikes. Your coop baby can deploy bombs, pushing items past spikes.
• Magnetism. If you have Magneto, you can clear fires, poop or laying bombs on marked rocks hidden past metal blocks and then attract the drops over to you. If you have the Bible, you can try and clear them anyway and see if, by any chance, the items dropped are good enough to activate temporary flight.
• Occasionally taking half damage after the Womb: if you have, say, just 5½ red hearts and a safe corner to park Isaac in, you can try and complete the room with your coop baby. If you manage, you're no worse than you were before. If you don't, the coop baby will die, but since it only had half a hitpoint to begin with, you won't lose additional health. In order to take full advantage of this you'll need a steady source of half red hearts such as Little C.H.A.D.
• Going beyond the health cap. Once you reach your health cap through excess of blue and black hearts and Dark Bum just keeps dropping more on the ground and you're about to leave the level anyway, you can transfer some over to your coop baby and pick the remaining hearts up. If your baby gets hit, oh well, you were going to leave those hearts behind anyway. If you get hit, you can reclaim those hearts merely by hitting Start.

All in all, this is rather situational and in many rooms there are no real safe spots to park one or the other player in. Still, it is IMHO good to know that, should the situation arise, you can take advantage from being your own coop buddy.