Learn Croatian: Njezin glas ima umirujući ton. - Her voice has a soothing tone.

It is a present active participle acting as an adjective. You form it by taking the 3rd person plural present tense of the verb umirivati (to soothe/calm), which is umiruju, and adding the suffix -ći.

Since umirujući is part of the object of the verb ima, why does it look like it is in the nominative case?

It is actually in the accusative case! It modifies the masculine noun ton. Because ton is an inanimate object (not alive), its accusative form is identical to its nominative form. The adjective umirujući simply matches this rule.

Could I use a relative clause instead of the participle umirujući here?

Yes, you could say Njezin glas ima ton koji umiruje (Her voice has a tone which soothes). However, using the participle as an adjective here (umirujući ton) is a very natural, concise, and elegant way to express this in Croatian.

Is there a difference between starting this sentence with Njezin versus Njen?

There is no difference in meaning; both mean 'her' and match the masculine noun glas. Njezin is generally considered the stricter standard form, while njen is slightly shorter and very common in everyday speech. Both are perfectly fine to use here.