GETS and IS are correct because it is related to the words teaching and showing.

Gets. 'teaching children.. ' is like a noun. What you teach them is not important. :)

".... GETS them excited." is correct. "Showing letters, words, etc. IS the best way." is correct. 'are' is wrong in that sentence because the verb is the action for 'showing' (and 'showing' is singular),, NOT for 'letters etc.' You can see if you remove the 'letters' part:: "Showing ---- is the best way (to learn)"

1. Gets them excited

2. Not sure what you mean ?

3. Are the best ways