不安と、病名がちゃんとあったことに少し安心して涙が止まらないーーー女性のこの日のツイートにはそう書かれていました。(source)
I don't know how to link "不安と" to "病名がちゃんとあったことに少し安心して涙が止まらない"(which function of と is it?)
And I don't understand why 少し is used if it makes her cry, it doesn't seem logic since if it is a weak emotion it shouldn't make her cry...? "I can't stop crying because I' m a little bit relieved"
と connects 不安 and 病名がちゃんとあったことに少し安心. It is a little off grammatically, but that's not too much of a surprise given that this is a tweet. Perhaps more proper way of saying this would be 不安と、病名がちゃんとあったことへの少しの安心とで、涙が止まらない
As for your question about 少し, if you read it like "a little bit of relief" resulted in tears streaming, that might come off as odd, but if you think of it as "finally I found that somebody understood this illness", which is what the name signifies, then hopefully you can understand why it made her cry. That sense of finding an answer gave her comfort.
She's describing this comfort as "a little" because knowing what the problem is still just the beginning, not the ending.
The sentence can be broken down as:
1. 不安と、
2. 病名がちゃんとあったことに少し安心して
涙が止まらない
The tears wouldn't stop because
1. worry (for realizing she had a medical trouble)
2. relief for knowing exactly what kind of medical condition she had.
