Learn Finnish: En ole ylpeä tästä tuloksesta. - I am not proud of this result.

This is because of adjective rection. The adjective ylpeä (proud) requires the thing you are proud of to be in the elative case (-sta/-stä). Since the noun is in the elative, the demonstrative pronoun pointing to it (tämä) must also match it and become tästä.

Since this is a negative sentence ("En ole..."), shouldn't the adjective ylpeä be in the partitive case (ylpeää)?

No, because it describes a specific person (I). When an adjective describes a human subject or a single whole countable object, it stays in the basic nominative case, even in a negative sentence. You generally only use the partitive for adjectives in negative sentences when describing uncountable things or abstract concepts (like "Vesi ei ole kylmää").

How do we get the word stem tulokse- from the basic dictionary word tulos?

Nouns ending in -os or -ös (which are often words formed from verbs) change their ending to -okse- or -ökse- when you add a case ending. Therefore, tulos becomes the stem tulokse-, and then we add the elative ending -sta.

Both words are in the same case, so why does tästä end in -stä while tuloksesta ends in -sta?