I think 'Hang out' and 'Hang with' are similar, there is no huge difference between these two words but 'together with' is different , because you can use it also as 'if you put two or more things together, you combine them to form a single thing. together with: Mix the flour together with the milk. together with something in addition to something else Your sandwich, together with Carl's salad, came to £8.50. Thank you

Hello, 'Hang Out' is a more commonly used American word but I will try to explain. I think 'Hang Out With' can be used in both senses you mentioned, in the house and also outside of the house. I could say 'Do you want to hang out at mine later?' and I could also say 'Do you want to hang out at the beach tomorrow?' When it come to 'Hang With' it can also mean the same, 'Do you want to hang with me later at mine?' and outside of the home could be 'Do you want to hang with me and Sean later at the park?' The last one 'Together With' can be like you said relate to two people who are dating or in love. I believe that phrases like these can always have a change of meaning depending on the context that the words or phrases are written in. I hope this helps.