Mifare 1k: Authentication Key A / Key B Blocks and Sectors

Question

I have to following Problem with the 1K Mifare Tag and ACR122U:

• First:

  Am i right, when i understand the Mifare Block Scheme like that:

  BLOCKS: &H0, &H1, &H2, &H3 --> Form Sector 1, where &H0 is the manufacturer block and &H3 is the block where KEY A and KEY B is stored?
  BLOCKS: &H4, &H5, &H6, &H7 --> Form Sector 2, where &H7 is the key storage
  
  EDIT: 13.08.16: i've found the answer for this particular part at: https://stackoverflow.com/a/28255795/6706324
  But in every article its said: using key a or key b - why "OR" and what about the following:

  What happens, if i change Key A and Key B to different values?

  I guess that then I can only access Bytes 1-6 in Block &H1 when authenticating with "&H60 -> key A"??

  Wondering, that with the manufacturer key &HFF, &HFF,..... i can access all 16 bytes when authenticating with key a.

• Second:

  Everything works well with manufacturer Key (FF..) when writing 16 Byte to &H1 and 16 Bytes to &H2 and then Authenticating with standard key A: &HFF &HFF &HFF &HFF &HFF &HFF --> i can read both sides of the 16 bytes (key a and key b side?)

  Question here: Do i have to authenticate Block &H1 with Key A, then write to it - and in a second procedure authenticating again to Block &H2 and then writing, or is just authing one block of a section necessary?

• Third: Everything stops working, when i do the following:

  1) Authenticating &H1 (because authenticating &H3 does not work), then writing the following in block &H3:
  
  Key A: &H31 &H31 &H31 &H31 &H31 &H31
  Conditions: &HFC &H3F &H0 &H69
  Key B: &H32 &H32 &H32 &H32 &H32 &H32
  

  Now when trying to authenticate Block &H1 or Block &H2 again, error "wrong key" occurs. I already tried that with Block 4,5 as DATA Blocks and 7 as Key Block

I've read so much information, but i don't find a good entry in this topic... thank you in advance