"I'm really sorry that I am carrying on with this"
You really never need to say sorry to us, or anyone else for talking about these things. You need to and that is that. Here at least you are also very welcome to talk about those things
" but, my counselor doesn't believe I'm telling the truth. He thinks I am overreacting so that I can be heard or get attention. This literally broke me for all of today"
You need a different counsellor. You actually need to be heard and you deserve to have some attention from someone to help with the problems you are facing. It sounds to me like your counsellor is not helping by invalidating how you feel. Your feelings are important, they are not overreactions, they are your reactions.. they are what they are and I do not think any of us get to choose how we feel. We can only choose what to do about it when we realise how we truly feel.
"a teenager so I cannot do anything by myself"
I think it depends on exactly how old you are, maybe if you are 16 you can see a GP in private without your parents knowing and then you could ask that doctor for a referral to a proper mental health service and get a proper assessment instead of being dismissed. I think that would be a good idea. If you are under 16 I am not sure. Does anyone here know the rules about this? And do you know any adult in your life who could potentially support you through this a bit.. a nice teacher perhaps? Someone trustworthy.
"my family doesn't support this but I do need help."
This is a shame because you really need their support. Have you tried writing down what you want about more support. Sometimes writing it can make it clearer because conversations tend to run away with themselves.
" Luckily, my mood has gone up significantly now and that is why I am writing here. However, I know that I am Just happy for no real reason so it is probably an empty feeling which will make me hate myself, even more, tomorrow if I do something like being nice to people I don't get along with etc. "
It sounds like your mood going up and down so much is really stressful. I have that a lot too and it is also utterly exhausting.
"What should I do? I no longer care about talking about my emotions and just want an answer. I know it won't necessarily be the best solution but I don't have any options left. Can I ask my counselor to refer me to someone who could help with this problem without my mum being involved?"
I would definitely ask. If your counsellor says no I would just go to a doctor and ask directly for yourself. You need, want and deserve to have more support than you are getting.
"Thank you in advance and once again, sorry for going on for too long."
You have not gone on too long. Just the right amount of time to let us know how it is for you. Which is good. Don't let the people you have in your life right now make you feel like you shouldn't be talking about your feelings. Every human being on this planet has the right to feel what they feel and to talk about it with someone who will listen. It is totally cool