Yes! In Portuguese you can use "the" in front of possessive adjectives.
It can but it is redundant since "nosso" already specifies which horse (the whole point of using the definite article "o" is to imply specificity).
Usually you will want to avoid redundancy while writing formal Portuguese (but for informal conversations/texts you'll be fine).
Agreed should be an option. Also annoying when the word is in there twice, once with and without the capital letter.
I understand you start a sentence with a capital letter, but they could just do that when you press check.