Is it appropriate to cold email somebody using their firm email on the internet if you are specifically interested in their work?
You can do this but I'd keep the email short and to the point in terms of why you are reaching out to them - don't tell them your life story, just explain why you are contacting them specifically.
And don't send lots of cold emails to people in the same firm - this gets around pretty quickly, even in large firms.
And finally don't expect a response if you do cold email - very much leave it at that if you don't hear back (I personally don't recommend chasing up or reminding them of your email if it is a cold connection).