It seems like negative consequences (punishment) isn't working, so it might be useful to try a positive program, as powerful as you can make it, if you can get the teachers on board.
Just off the top of my head, maybe a token program might work for her. She could earn tokens or stickers for each occasion of "positive" behavior: playing well with another student, sharing, etc... whatever behaviors are the opposite of what she is doing that you don't like (e.g. the opposite of bullying is probably cooperative play). Or it could be a program where she earns a sticker or token for each hour during school that she doesn't display the problematic behaviors. Then she could exchange the tokens/stickers for something desirable (a small toy, an outing with you, etc).
If she is only "bullying" once a day, and only occasionally takes something home with her that isn't hers (like once a week or so) then perhaps you could ask the teacher to send a report home each day, and with 5 "good" reports she gets to choose a fun activity with you over the weekend.
Some degree of bossiness is really common among 5 year olds, and maybe she is really just being bossy. Its also common at that age for 5 year olds to start with the "you can't play with us" and "you're not my friend anymore; Kim is" which needs to be addressed, of course, but also you shouldn't consider it too worrisome. Rather than a timeout, I would think it would be better for the teachers to simply restate the rule ("everyone can play" or "hurtful words are against the rules" with an explanation of the words she used that were hurtful). If the bullying is more extreme, such as pushing, shoving, hitting, or coersion, then a time out might be a good solution.
Lying is pretty common too and shouldn't be worrisome. Lying at that age is usually about expressing what they wish was true, and doesn't involve the same level of deceit and manipulation that adults think of when they think of lying. She wishes her day was fabulous. She wishes you believed her day was fabulous, so that she would not be punished. She wishes the teacher had given her permission to take something home.
Taking things home could be several things. Could be that she really liked that particular toy and wanted to take it home. If that is the case, can the teachers allow her to choose from some toys a toy to borrow for the evening, with the knowledge that they will be returned in the morning? Taking things home could also be anger at her teachers and a desire for retribution, but she is afraid to act out directly because she knows she will get in trouble. Taking things home could also indicate she loves school and wants to take a part of it home with her (but I doubt this is the case. If it is, perhaps the teachers could draw a special card or picture for her to take home each day).
But if it were my DD, I would probably pull her from school. It sounds like something about the school environment is causing her stress. Maybe its just a bad match between her and the teachers. Maybe there are some other kids that are causing her distress. Whatever it is, I think I would take my DD out and see if the behaviors resolved at home. You might not be in a position to do this, though.