Learn Russian: Она выглядит усталой.

After the verb выглядеть (to look / appear), we usually use the Instrumental case to describe how someone looks. It indicates a temporary state, condition, or impression. You might sometimes hear the Nominative (усталая) in casual speech, but the Instrumental (усталой) is the standard way to express this.

Does the verb "выглядеть" conjugate like a regular "-еть" verb?

Actually, it conjugates like a group 2 ("-ить") verb, even though it ends in "-еть". It comes from the verb глядеть (to look/glance), which is an exception that belongs to that second group. Also, the prefix вы- always pulls the stress to itself, making the form we see here она выглядит.

Can I use "смотреть" instead of "выглядеть" to say "she looks tired"?

No, смотреть means "to look at" or "to watch" something actively (like watching TV or looking at a picture). Выглядеть means "to look" in the sense of "to appear" or "to have a certain appearance". Since we are talking about her appearance, you must use выглядеть.

What is the dictionary form of the adjective for "tired"?

The dictionary form (masculine Nominative) is усталый. Because the subject is Она (she), a feminine ending is needed, and because of the verb выглядит, it shifts to the feminine Instrumental ending -ой, making the final word усталой.