How to join and 'multiply' two lists?

RememberOuter everytime you try to do something you call as 'multiply'.

Outer[Join, A, B, 1]

{{{1, 2, 3, 1, 2}, {1, 2, 3, 3, 4}, {1, 2, 3, 4, 5}}, {{1, 0, 3, 1, 2}, {1, 0, 3, 3, 4}, {1, 0, 3, 4, 5}}}

Join as the function, and A and B as two vectors, then do Outer at level 1 shall give you a proper result.

I don't know what you want for the second part of your question, but I suppose adding a filter directly to A will do the job:

Select[A,#[[2]]!=2&]

This appears to be 7-10 times faster than Outer, depending of lists length:

 With[{l = Length[B]},
   Function[
      subA, Join[ConstantArray[subA, l], B, 2]
   ] /@ A
 ]

Join @@@ Tuples[{A, B}]

{{1, 2, 3, 1, 2}, {1, 2, 3, 3, 4}, {1, 2, 3, 4, 5}, {1, 0, 3, 1, 2}, {1, 0, 3, 3, 4}, {1, 0, 3, 4, 5}}

Distribute[{A, B}, List, List, List, Join]

{{1, 2, 3, 1, 2}, {1, 2, 3, 3, 4}, {1, 2, 3, 4, 5}, {1, 0, 3, 1, 2}, {1, 0, 3, 3, 4}, {1, 0, 3, 4, 5}}

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