HindIII and HaeII are proteins known as restriction enzymes. Restriction enzymes are found naturally in bacterial cells and their function is to cut up foreign DNA (such as viral DNA) at a specific target sequence. These target sequences are usually (although not always) what is referred to as “palindromic”. A palindrome is something that can be read the same way both forwards and backwards. You may have heard of palindromic words, such as racecar, mom, or civic. These words can be spelled the same way both forwards and backwards. A genetic palindrome, however, is slightly different. A molecule of DNA has a specific orientation based on the chemical structure of the molecules that make up DNA. A molecule of DNA is composed of two strands, each strand complementary to each other and running in the opposite orientation. We refer to the direction of the strand as either being 5’- 3’ or 3’- 5’ (this is determined by the direction the sugar backbone of DNA is facing).

Complementation in DNA refers to what bases are paired together in each of the two strands. The four bases of DNA are A, T, G, and C, where A and T pair together and G and C pair together. Now that we know a bit about the structure of DNA, we can discuss a genetic palindrome. When we “read” DNA, we always read in the 5’- 3’ direction, just like you would always read left to right on a page. Let’s look at the HindIII recognition sequence as an example:

5’ AAGCTT 3’
3’ TTCGAA 5’

You may notice if you read the top strand in the 5’- 3’ direction and the bottom strand in the 5’ - 3’ direction that the sequences are both AAGCTT. This is what is meant by a genetic palindrome. A sequence that may be considered palindromic as a word, such as AAGGAA, would not be considered a genetic palindrome. If you look at this sequence and its complement:

5’ AAGGAA 3’
3’ TTCCTT 5’

You can see that the 5’- 3’ sequence AAGGAA is not the same as the 5’- 3’ sequence TTCCTT. Therefore this is not a genetic palindrome. To answer you second question, if you want to pick a restriction enzyme that will cut your DNA into to small pieces, you would pick a restriction enzyme with a shorter recognition sequence (4 base pairs instead of 8 base pairs). The reason for this is that the probability of a 4 base pair sequence randomly showing up in DNA is much more likely than an 8 base pair sequence randomly showing up. For each position in the DNA, there are 4 possible base pairs (because there are 4 different DNA bases, A,T,G,C). If we think about the possible permutations for a 4 base pair sequence it would be 4 possibilities for the first base, 4 possibilities for the second base, 4 possibilities for the third base, and 4 possibilities for the fourth base.

Mathematically this would be:
4 X 4 X 4 X 4 = 44= 256 possible sequences Only 1 of those sequences is the correct recognition sequence, so there is a 1/256 chance of finding our specific sequence in the genome.

Now, if we were looking for a specific 8 base pair sequence, the possible permutations for 8 bases would be:
4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 = 48 = 65,536 possible sequences

Therefore there is a 1/65,536 chance of finding that specific sequence. This is much less probable so we will find many fewer specific 8 base pair sequences than 4 base pair sequences, which means the DNA will be cut less if your recognition sequence is 8 base pairs instead of 4 base pairs.

I don't know enough to recognize specific codes like that. I will note that AAGCTT and AATT both are mirror images of each-other; the complementary strand of DNA will have the same sequence (read backwards, switching A for T and C for G, and you will see what I mean).