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My tendency is to look at every little detail. This allows me to relax knowing I've got things as good as they can be. This approach might not suit people who get anxious about details and prefer to "wing" a few things here and there, but being thorough has always worked best for me.
>-How to not make ymself look like a weird twat on camera (what do I put in the background, do I still wear a full suit etc)?
Test run everything, a day before if you have to. Check how you look in your webcam. Check the lighting, environment and microphone, using the exact same hardware and room you'll take the interview in. Call a friend first and ask if you're coming through clearly, the picture is okay, and you're looking generally employable. A clean and professional outfit never counts against you.
>-How do I make a good impression when other candidates will be there in person?
Preparation. Doing your background research on the job, the organisation, the team, etc., and keeping a clearly worded motivation in mind.
Think it through as though it were really happening -- not just a prospective interview, but going from point A (where you are) to point B (working and living with them). Have you visited that part of the world before? Do you know what the culture is like? What elements of the job would you like to know about, but weren't able to find from your research?
Think about the interview from their perspective: what would they realistically be looking for from you? No one expects a perfect candidate, but what could they reasonably hope for? What would you look for if you were Skyping a candidate on the other side of the world? Is it a small team or a big corporate office? The more you can anticipate their questions and desired traits, the better you can prepare.
>-Is it weird if I stay at work and do it and explain to them that it's because my home internet is shit?
Interview where you feel most comfortable -- you don't need to explain anything. The most important thing is having an attractive and distraction-free environment that you enjoy talking in (or at the very least, can think clearly in).
You mentioned it's a niche job -- that probably means that it's a tightly-knit(ish) industry, where people will know or have heard of others. If you have any other contacts at all working in this area, leverage that now and ask if you can talk to them (or better yet, Skype them) about this particular place where you have an interview lined up. The more you get used to talking about the relevant work, the more thought you put into it, the better you're able to communicate it. It'll all show on the day.
There's lots of other generic tip lists out there, but it really boils down to just doing your preparation and presenting yourself well. The Skype is just an added element to consider that comes to "make sure everything works and close your other shit first".