Learn Finnish: Tämä koira on minulle hyvin rakas.

This ties directly into adjective rection. The adjective rakas (dear) expresses an attitude or feeling directed toward someone. In Finnish, adjectives of interpersonal attitude require the target person to be in the allative case (-lle). Here, it translates perfectly to the English structure 'dear to me'.

Could I use the genitive case and say minun hyvin rakas instead?

Not in this exact sentence structure. If you want to say 'my dear dog', you would use the genitive for possession: minun rakas koirani. However, in this sentence, rakas is a predicative adjective describing the subject. Because it functions independently here, it dictates its own case requirement (the allative minulle) to show who holds the feeling.

Why is the adjective rakas in the nominative case rather than the partitive, since it deals with feelings?

When using the verb olla (to be), the case of the descriptive adjective depends on the subject's grammatical nature, not the emotional meaning. Because the subject Tämä koira is a singular, countable noun, the adjective must agree in the nominative case (rakas). You would only use the partitive if the subject were an uncountable mass noun, like food or water.

I usually see tosi or erittäin for 'very'. Does using hyvin change the tone of this sentence?