I noticed that in several places in the tmux panes and menus, there are chartacters being displayed in the boundaries.
Example 1:

How do I get rid of these characters and make them solid colored boundaries with nothing in them!
These are ACS line drawing characters. Putty does not support ACS line drawing in UTF-8 mode. For this reason (since tmux 1.4) tmux uses UTF-8 line drawing characters when UTF-8 is enabled. Make sure tmux knows the terminal supports UTF-8, either by using the -u flag (tmux -u new and tmux -u attach) or by setting LANG, LC_ALL, LC_CTYPE appropriately.
Alternatively, you may be able to configure Putty to disable its UTF-8 support, but I don't know how to do this.