Learn Finnish: Heitä lumipallo tänne! - Throw the snowball here!

In Finnish, spatial adverbs change depending on movement. Because the snowball is moving to a place, you must use the directional form tänne (to here). The word täällä just means "here" when there is no movement.

Could I use tähän instead of tänne?

Yes, but the meaning changes slightly. Both mean "to here," but tänne refers to a broader, general area around the speaker. tähän refers to a specific, exact spot. Since you usually throw a snowball to a general area, tänne sounds very natural here.

Why doesn't the object have an -n ending here (like lumipallon)?

This sentence is a command. When you use an imperative verb like heitä, a total object takes the basic dictionary form instead of the usual -n ending. Therefore, we say lumipallo instead of lumipallon.

What is the dictionary form of heitä, and why does it only have one t?

The dictionary form is heittää (to throw). The command form is built using the same strong-to-weak consonant gradation as the "I" form (minä heitän). The double tt weakens to a single t.