It's a fair question and an interesting one:
the most common thing is to discuss each-other's location.
Very briefly, to establish a setting.
So, "your city" issues:
weather...
"I've been" issues...
The absolutely typical opening lines, in US or UK remote business conversations are:
Hi, I'm John
HI, I'm Steve and this is Jane
I believe you're in Chicago is that right?
Yes, you're in Shanghai right?
I've visited Chicago a few times when I worked for XYZ. My cousin lives near Chicago.
I've never been to Shanghai, but I am often in Hong Kong.
I hear it's cold this time of year, same in Shanghai.
So, regarding the project, any files I should look at or ...
I would almost say, it is de-rigeur to establish what city everyone is in.
So simple "your city" conversation such as
You're in Atlanta right, same time zone as New York?
Actually we're an hour west of New York. You're on Paris time. Let's get started then.
Funnily enough I lived in Paris for three years as a child, I don't speak much French though.
Don't forget to let the "important person" lead the conversation.
When you first connect, simply say I'm Jane, Nice to meet you and then let them talk.
Very often the "second" thing you say is "My camera's working ok for you, correct picture?" or "Sound quality is OK for you, should I use my microphone?" and that naturally leads to you actually beginning.
So!
simply say "I'm Jane nice to meet you"
it is extremely common to have >short< comments on each person's city. the most common is "I have/not been there", "Question about weather"
"My microphone is working ok?"
and then, the important person says: "Let's get started .." or You say something like "So, what would you like me to do first today on the call..."
As a matter of fact, your example ... It's a very cold day in New York, How is your place? is perfectly correct.
Thanks for interviewing me today. I believe you are in New York - it must be cold today?
Yes cold day. You're in Kyoto, I've only been there briefly on holiday.
Ah, yes I have spent two weeks in New York when I was a young man. Thanks again for calling, how may I serve you today, what should we look at first?
Perfect modern Skype etiquette.