Learn Hardcore Finnish: Minä odotan vastausta koko ajan. - I am waiting for an answer the whole time.

In Finnish the verb ending already shows person and number, so Minä (“I”) is optional. You can simply say
Odotan vastausta koko ajan.
Dropping Minä is perfectly natural in conversation. Including it adds clarity or emphasis: Minä odotan… stresses “I (as opposed to someone else) am waiting.”

Why is vastausta in the partitive case instead of nominative vastaus?

Verbs that describe expecting or waiting for something—like odottaa—take a partitive object when the action is incomplete or ongoing. Since you haven’t received the answer yet, it’s an unfinished event, so you use the partitive singular vastausta (from vastaus, “answer”).

Could I use the accusative vastauksen or nominative vastaus here?

No. With odottaa, the object must remain in the partitive because the awaited thing isn’t a completed, countable whole. Using vastauksen or vastaus would be ungrammatical in this context.

What does koko ajan mean, and how is it formed?

koko ajan literally means “whole time” and translates as “the whole time” or “all the time.” It’s built from:
koko = whole
ajan = genitive singular of aika (“time”)
Because koko requires its noun to be in the genitive, you get koko ajan.

Why is koko ajan placed at the end of the sentence? Can I move it elsewhere?

Finnish word order is quite flexible. The default is Subject–Verb–Object–Adverbial, so putting koko ajan at the end is perfectly normal. You could also say:
Koko ajan odotan vastausta.
or
Minä odotan koko ajan vastausta.
Moving it changes the focus slightly but doesn’t break grammar.

How is the verb odotan formed from the infinitive odottaa?

odottaa is a Type I verb. To form the present tense 1 sg, you take the infinitive stem odota-, drop the -a, and add the ending -n:
odota- + n → odotan
The full paradigm is odotan, odotat, odottaa, odotamme, odotatte, odottavat.

Can I say “I have been waiting for the answer the whole time” using a perfect tense?

Finnish often prefers the simple present for ongoing actions: Odotan vastausta koko ajan.
If you really want a perfect nuance, you can say:
Olen odottanut vastausta koko ajan.
Here olen odottanut is the present perfect (“have been waiting”), but it’s less common in everyday speech for this kind of continuous waiting.

What’s the difference between aina and koko ajan? Could I use aina instead?

aina means “always” in the sense of a habitual or repeated action (“I always…”).
koko ajan emphasizes continuous duration during a specific period (“I have been waiting all the time”).
If you say Odotan aina vastausta, it means “I always wait for an answer” (as a habit). To stress nonstop waiting right now, use koko ajan.