Deciphering “easy” ciphers without hints
I had a look at classic transpositioning and substitution methods which lead me to encryptions like Vigenére, Autokey, Beaufort and so on. However, those are designed to work with non-numeric alphabets as far as my understanding goes.


While most classical ciphers are applied to the usual alphabet (without different cases), they are not limited to that. It's perfectly fine to have exactly that alphabet you mentioned. In your cryptanalysis you will most likely use number representations of characters anyway (to use addition and modulo), then it just comes down to ordering the alphabet.


  About the cipher: The Cipher contains [a-zA-Z0123456+] without whitespaces, while some patterns are recurring (same characters are repeating in the same order a few times in the cipher).


If there is structure like this, it's very likely to be Vigenere or something similar, operating in fixed "blocks". The distances from the repeating patterns will indicate the block size, as they are multiples of the blocksize.


  I tried a frequency analysis to see if it's just substituted monoalphabetically sign for sign - without success (many results staying at ~8%, no one above 10%)


If you have the symbols a - z, A - Z, 0 - 6, how can  58 symbols have "many at 8%"? Still, it looks like Vigenere fits.

A few things to consider:


Having different cases can imply, they have an alphabet of 2 * 26 (and then add the numbers), or they just use the case of the plaintext, but that should be quite obvious (the distribution is more even in the first case). Alternatively they could just have used a single alphabet and randomly used upper or lower case.
Focus on finding out the block size first. You mentioned index of coincidence, but I guess you meant the "Friedman test". And you have to keep in mind, that your alphabet might be different than the standard one. Use the actual index of coincidence to find out the block size. As an algorithm, you can also calculate the autocorrelation directly (for shift $i=1,2,\dots$, count the positions where the text and the shifted text have the same symbol). The correct block size (and its multiples) will be a outlier(s) from what you would get from a uniform distributed alphabet (for all the wrong block sizes).


From there on, it's just frequency analysis for each position in the block.