Direct answer

The following formula return the expected results for the three cases exposed in the question. It should be added to cell A2 and cells in A3:A13 should be empty.

=ArrayFormula(
  IF(
    NOT(
      (ROW(B2:B13)-1=1)
      *
      (B2:B13="$")
    )
    *
    (
      (B2:B13=">")
      +
      (B2:B13="$")
      +
      (ROW(B2:B13)-1>=Vlookup("$",{B2:B12,ROW(B2:B12)},2,0))
    )
    *
    NOT((B1:B12=">")
    *(B2:B13="$")),
    ".",
    " "
  )
)

Explanation

In this case the use of ArrayFormula function together with references to B2:B13 will return an array of 12 rows and 1 columns with the expected values.

The comparison of a range or array with a scalar value returns an array of the same size of the range or array. All the tests should return an array of the same size.

AND and OR functions could not work with arrays, but Google Sheets interpret TRUE as 1 and FALSE as 0, so, instead of using AND function use * and instead of OR use +.

As + and * take precedence over comparison operators each test should be enclosed in parentheses.

Please try:

=if(COUNTIFS('Our Animals'!$B:$B,A2,'Our Animals'!$C:$C,"no")=0,"Yes","No")  

in Animal Types B2, copied down.