Hi K.,
Long answer ahead ...
In no particular order:
1. Is your fiance living in the house with you? If so, he has just as much right as anyone to expect the kids to behave in his house. It is not question of the marriage license being signed or not - he is an adult.
2. I'm sure you already know, you cannot make your son respect you by just telling him to do so. Make hard and fast rules and stick to them. It sounds to me like he is acting up because he needs direction and discipline; but he is running the show. That must be scary for him too. You don't have to be "mean" to him, just let him know that you are the parent and it's your house and he has to abide by the rules ... or else.
3. Are your children aware of your issues with depression? I don't think that they are old enough to appreciate what you are going through and your 13 year old may see this as a sign of weakness.
4. You mentioned -- twice -- that your son worships his dad. It seems to bother you quite a bit. Have you talked to your ex about this? If you two are truly working together with the kids then he needs to start demanding good behavior and respect towards you as well.
Are you the parent that the kids live with all the time? My parents were divorced and I lived with my mother so my dad seemed like the ultimate fun guy. He didn't see us very often so he made sure we had a good time. Mom had to enforce the rules and I know we would be bad to her when we first came back from a visit with dad. It must be rough but now that I am older I totally understand and respect what she did for us kids.
5. Most kids tune their parents out when they are saying something that they don't want to hear so you'll need to repeat yourself often. Don't give up! As annoying as it is you have to keep on them. Punishment is a great deterrent as well. Tell your son that he has to take out the garbage -- repeat that until you are sure he has heard you. Also tell him what the consequences are if it is not done before he goes to bed (or whatever the time frame is). For instance, after the 5th time you finally got his attention. Son, if you don't take the garbage out before bed-time tonight you will forfeit your video games for the rest of the week. Then, walk away. No reminders for him. He probably won't do it the first time because he's not used to you following through so it is extremely important that you do not back down. Take his games away for the whole time that you said you would. Don't give him a punishment that you can't stick to. Sooner or later, he'll start paying attention.
6. Chores are basically the same thing but I would recommend a chart. Put your son's chores and your daughter's chores on a board and keep track of them throughout the week. Do they get an allowance? They shouldn't get one unless all of the chores are done that week. Alternate responsibilities too so that one doesn't think you are favoring the other.
7. Your son is bullying his sister for at least two reasons: a) she is younger and has less chance to defend herself and, b) he feels that she is the recipient of more affections then he is and this is his way of getting some attention too. When he does this to her, don't run to your daughter and hug her and yell at your son for knowing better. Instead, separate them immediately. Let each of them cool down before addressing the problem. Then get all 3 of you together in the room to discuss why it happened. Talk to your son first. Tell him that you love him but explain why he cannot continue to hit his sister - or whatever. Ask him why he wants to hurt her. Tell him how it is effecting the entire family. Then ask your daughter to tell him why she doesn't like it. That should open the door to more communication.
8. If you stick to your guns, your problem should subside (remember, he's still a teenager so you will have disputes with him for the next 7 or 8 years!). However, as a last resort, ask your ex to take your son for 6 months or so. It will hurt but if you've exhausted every other avenue (like I mentioned above and, of course, your own ideas too) then maybe you sit your son down with your ex and say, "Son, your dad and I agreed that you should spend some time with him. It seems that you would like to do that anyway and we cannot continue to have disruptions in the house. This is a temporary situation and I am looking forward to having you back with us full-time but for now we all just need a break." Making your ex share in the discipline on a more full time basis may help your son to realize that no one walks on water.
Good luck and let me know how you're doing.
K.