Learn Spanish (Latin America): El perro es leal y feliz.

No, they stay exactly the same: la perra es leal y feliz. Unlike adjectives that end in -o or -a, adjectives that end in a consonant (like leal and feliz) or the letter -e don't change based on gender. They use the same form for both masculine and feminine nouns.

I learned that emotions use está instead of es. Why does this sentence use es with feliz?

You're right that está feliz is used to say someone is happy right now (a temporary mood or state). However, using es (from the verb ser) describes a permanent characteristic or personality trait. Here, es feliz means the dog has a generally happy nature, not just that it happens to be in a good mood at this exact moment.

How would these adjectives change if we were talking about multiple dogs (los perros)?

They would both become plural, but you have to watch out for a special spelling rule. To make leal plural, you just add -es to make leales. But for feliz, Spanish words ending in -z change the z to a c before adding -es. So it becomes felices. The full plural sentence would be: Los perros son leales y felices.