Is there a formal custody agreement?   I have a feeling there is not.  That would be the first step in this situation.

I wonder if you are like I was.  I moved in with DH without really thinking thru the SKs visitation or what my part would be.  Our unspoken agreement was I could be a SAHM but I would have the SKs as much as he could get them which was a lot once  BM realized she coukd dump them off.  Sound familiar?

You have an 11 months old baby. Your SO works a lot and you are supposed to watch for SKid ? 
If you say anything, your SO starts an argument. Waou ! Of course you're not a nanny or house maid. 

If i were you, i would stop immediately to watch for Skid when your SO is not home. He must understand it's not a duty at all. Plus, the way his parents treat you won't help in the future. If the kid sees they disrespect you, he will do the same.

Stop to watch for the kid. Be really clear on that. When your SO would be ready to talk about it, then explain to him you are not the mother, not even an official parent and you don't appreciate that it is imposed to you. Explain that it's not a duty, and that you must agree, and that since he doesn't respect you, you don't want to watch for his kid untill he gets it. 
If he doesn't have time to care for his own kid, he is the one who has to find solution, and you are not part of the equation. 

"I rarely get time to myself even when he is home & If I do step out on my own there's always an argument"

Everybody needs time to decompress in the day. Make plans for yourself and follow them. He just can't be bothered taking care of any of his kids. Tuff on him. You are allowed to have a life outside of babysitting. Stand up for yourself.

"A couple weeks ago i was asked by him if I could watch his child so his ex could go to Miami with her friends even though she just came back from a California trip"

The fact that she just came back from a trip is of no importance. What is of importance is that you are under no obligation to take care of her kids. Either you say an emphatic "no" and stick to it or tell her your going rate and that you want to be paid uprfront or you won't be looking after her kids for her.

 "I feel like I am getting all these responsibilities of a wife but he won't make me one"

I think your idea of what a wife is is a bit skewed. A wedding band does not make you a doormat, a babysitter, a dogsbody. It makes you an equal partner in your relationship. This guy isn't interested in being an equal partner, he just wants an easy life and wants you to shut up and do whatever it is that he doesn't want to do. This is not marriage material.

Start looking for daycare for your little ones (yours only - what happens to his is entirely his and his ex's problem) so that you can get a job and plan your escape. This is not a life. Take a look over your past few posts and see how often his actions are designed to cut you off from the outside world and keep you in your place...

Also, stop trying to discuss anything with these people, just don't make yourself available to do their jobs for them. Start going out when his kids are due over. Let him scramble to find a solution or send them back to their mother. You are not their unpaid skivvy and you deserve a sight more respect than you're getting.

You are being used.

And I paid $750 for someone to watch my sox year old and that's in addition to the cost of summer camp during the day so that whoever is watching her can get a break

Ridiculous! Even if you WERE married, it's not an expectation that you should be responsible for HIS kids. Whatever "arrangement" he had with BM, that needs to be it. HIS arrangement. If he agrees to take SK, then HE needs to be the one watching them. 

How old is SK? If your SO continues to take advantage, I would do the following. When you're home alone with the kids, text SO "Myself and 11 m/o are running errands. We will be back later. Oh, your child is at home btw." Go about your life as you normally would. Go visit family or friends last minute. YOUR life shouldn't revolve around BMs schedule. Plain and simple.

SK is NOT your responsibility. If he pulls the "you don't like my kid" card, throw the "I don't like that you are spineless when it comes to your XW." It's a major turn off for me. A man that can't stand up for himself. Totally pathetic. 

If he can't see your side of this, I would dump and run. Good luck to him finding a GF that also wants to be a free nanny. 

This sounds frustrating.  Do you work and if not, do you see room in the budget for him to pay for childcare for his child if you can't or won't watch them?  If you don't work, would you go back to work so you could cover your half of your child's daycare needs (your SO covers the other half).. and contribute to the other household expenses so money is freed up to pay for childcare for the Skid when needed?

If you are working and contributing a fair share to living expenses in your home.. then he really needs to find another babysitting service because a job and an 11 mo old has to be exhausting.. (mind you he should be helping to care for both the kids after work.. so you get time to decompress).

But, I can see being frustrated when he agrees to extra time from his EX and then obligates you to do the extra work of watching his child.. maybe there could be some additinal help you get.. in childcare or homecare.. meals or maid service to give you a break?

You have an SO problem most of all. He's very happy with you carrying the entire parenting load for his children. So is BM. The fact that you are NOT happy is apparently unimportant to the person who is supposed to be your partner and love you above all else. Is this a good relationship for you? From what you've written here, it doesn't appear so but only you know that for sure. Not allowing you your own time, activities, friendships, etc. is a form of abuse..

I think I'd try one more time to have a rational conversation with him -- prefaced by a statement that you will not tolerate an argument or his anger about your legitimate feelings and concerns. Then if he does get angry, leave the room. Leave the house if you must. Do not tolerate his disrespect.

You can stop being the default babysitter, too. "No" is a perfectly fine word, and you can stop being available. SO and BM will ramp up their demands so you will have to be strong and not cave in to that.

And you want to marry this man why?

You are being disrespected and used.  I'm guessing from your comment about being with your baby constantly that you don't work outside the home.  If that's the case, it's a mistake.  It's risky to be a SAHM and rely on a boyfriend to support you.  I would start taking steps to find a job and a daycare arrangement for your child. Let your boyfriend and BM worry about the skid.  I don't see anything changing unless you force a change. 

If you are expected to act like a parent, you shoukd have an equal say on when the child comes outside of the custody time, and be a part of any conversations. If they don't want to give you that power, you shouldn't have the responsibility. Responsibility without control is misery. 

they need a babysitter/parent for their kid, and your SO needs a warm body in bed. He doesn't seem interested in either of his children and seems like a crappy parent and a crappy partner to you. Why would you "settle" for this man and marry him? This doesn't sound like a good situation. 

I hope you can tell him exactly what you NEED and WANT in your life, relationship and your home. If he doesn't work hard to make that happen, then you should assume there are things more important to him than you. And you should leave. If you don't have the means to leave, start working on that. 

The fact that you cannot bring up these fair, reasonable issues to discuss with him makes me so frustrated for you! Why are there so many men who cannot have an adult conversation?? He brought two children into this world, time to grow up and be the entire man!