Learn Italian: Che seccatura che è questa dogana! - What a nuisance this customs office is!

The first che acts like "what a" in English, forming the exclamation Che seccatura. The second che is the pleonastic (or "extra") word you are learning in this lesson. It acts as a bridge connecting the exclamation to the rest of the clause to add exasperation and emphasis. It doesn't translate into English, but it makes the sentence sound much more natural and spoken.

Can I remove the second che and still keep the verb è?

You generally wouldn't. If you drop the second che, it sounds much more natural to also drop the verb and just use a comma: Che seccatura, questa dogana! Using the verb without the bridging che makes the exclamation sound clunky. The pleonastic che is exactly what allows you to include the verb è smoothly while maintaining the emotional punch.

Why is the subject questa dogana at the very end instead of before the verb?

In emphatic exclamations like this one, the emotional reaction (Che seccatura) is the most important part of the sentence, so it comes first. When you use the pleonastic che structure, it is standard practice to place the subject right at the end of the sentence as an afterthought or clarification.

What exactly does dogana refer to here?

Dogana means "customs". It refers both to the physical border checkpoint or airport desk and to the bureaucratic process itself. In this context, the speaker is complaining about the entire tedious experience of getting through it.