Question about JFLAP and automata

My first question is... In JFLAP there is a square symbol when building a turing machine. What does that symbol stand for? Does it stand for empty string?

My second question is... for a turing machine that reads for example ####abbcb#### and when read all symbols give ####bbcb#### then the turing machine doesn't need to read all symbols, right? Only to the last b? Or does it need to read all symbols to be 'valid' for that input string?

Last but not least I would appreciate if someone could take a look at the following turing machines and automata's and let me know if I understood it correcly and if something is wrong please let me know.

Indata ####abbcb####

Outdata ####a####

http://img69.imageshack.us/f/uppgift4ajff.png/

Indata ####abbcb####

Outdata ####bbcb####

http://img17.imageshack.us/f/uppgift4bjff.png/

1. a(bb)*a*

http://img545.imageshack.us/f/uppgift21jff.png/

2. a(bb)*b*

http://img823.imageshack.us/i/uppgift22jff.png/

3. a*aab*bb

http://img811.imageshack.us/i/uppgift23jff.png/

4. Σ = {a,b,c}, string "abc" not being allowed

http://img861.imageshack.us/f/uppgift2bjff.png/

Thanks!