A little backstory: I came out to my husband as poly about two years ago. We have been married five years, and though I've had a few casual relationships, I have been in a serious relationship for about five months now and put all other casual relationships aside for the time being. My husband and I have both known my current partner as a friend for many years.
We are all in our late thirties, and my husband and I have a small child together. The three of us all spend time together, and I spend 1-2 nights at my partner's house. He does not stay over at our house because of his pets.
That's convenient, that your friendship became a romance, so your kid was already used to this guy being around the house. It may not be necessary to explain the romantic part at all when the kid is only 8-9 years old. But once they hit the pre-teen years, they might start to ask questions.
We have been considering the idea of all sharing a home together in the future.
What is your sleeping arrangement? Do you have separate bedrooms or all share a bedroom? If your two partners each have their own bedroom, how do you split up your nights?
I am a hinge with 2 partners. Both of my partners also have one other partner each.
My gf spends Tuesday-Friday with me in our shared home. My bf spends Friday night-Sunday evening with me, although sometimes he pops over for a few hours on Monday, but doesn't spend the night. Therefore, I only ever sleep with one partner at a time. I get my free day from Sunday night-Tuesday evening. My gf is usually back here for dinner on Tuesday.
My gf spends her long weekend at her bf's house. He doesn't have any other partners.
My bf has one other partner, but she's a bit long distance, about an hour away, and she's married to a man, with their adult daughter living with them. It's a fairly new relationship (since about July, I think). They've never been able to spend the night together, as neither of them can host for that.
So, we each have our own arrangements, to give you a range of ideas.
If you're married, how do you ensure your partner doesn't feel "left out?" Both legally (medical, etc) and emotionally?
None of us are married, except for my bf's gf. So far, I think they split paying for their dates outside the home, but she often has him over for a homecooked meal with her family, and they just stay in, watch a movie, play videogames. I guess since he has to pay for his gas to get to her, a homecooked meal is a fair exchange!
Since I've been with my gf for 15 years, and my bf for 3, I considered them my spouses, even though we aren't legally married. Sometimes I wish I were married to my gf, but that wouldn't be fair to our bfs. Besides, I was married for about 30 years, and there were downsides to being married that I didn't love. I like my autonomy.
My gf and I share most of our expenses (rent, utilities, food), since we both make about the same amount of money. Her bf is much more well off than either of us, so he helps her out with major purchases, and she is very appreciative of that. My bf pays for his share of the food he eats when he's here, and I cook most of the meals (although he helps, and makes his own breakfast usually). Otherwise, he's more about the "sweat equity," doing heavy yard work, housework and repairs, which would be too much for my gf and me, so we don't have to hire handymen or yard work guys to do the jobs he's taken over!

He also does daily jobs like dishes, carrying in groceries, laundry and vacuuming.
How do you explain your living situation to your child?
All my children were young adults when their dad and I split. (We'd been monogamous during their early childhood.) I just simply explained I was going to be dating polyamorously, and since they were raised in a progressive environment overall, they seemed to accept it okay after seeing how it was going for me. It didn't impinge on their lives too much, because they were teens/early 20s at that point, and more interested in their own lives then what choices Mom was making.
I introduced my gf to them pretty early on, and they all loved her right away. I kept any men I was dating more separate from them until I'd been dating them for a while. Then I kept their interactions fairly short.
It can be tricky to explain who this new adult is in their parents' lives to younger kids, especially if you're not "out" as poly in your community/with your extended family. We have threads on poly and kids, so you can do a search on how it has gone for many people. Generally, I have seen that it is like any time a parent divorces. Mom or Dad are going to be dating. A lot of people don't introduce a new dating partner to their young children until they've determined that that person is a "keeper," so that the kids don't start to get attached to the person, only to then lose them if it doesn't work out.