Emails and conference calls while in the USA under ESTA (B-1 Waiver)

Look at the end of your quote:

directly connected with and part of your work abroad

Emails and Skypes with your foreign employer and foreign colleagues meet that test. Emails and Skypes with US clients of your foreign employer might or might not. Emails and Skypes with US entities who are giving money to you personally and directly definitely will not. It will be local employment, which is forbidden.

The issue, as always, is with the word work. We think we know what it means. We do not. It is a jargon word for customs officers with a different meaning than the dictionary or everyday conversation meaning (as it is for physicists in some circumstances.) What you describe is business and is allowed. You can't know this by understanding the everyday meaning of work or business but only by reading their documentation. When they say work they mean local employment and when they say business they refer to things directly connected with and part of your work abroad (and yes, I know that phrase has work in it, so be it. Employment would be better there, for sure.)

Tracing down the legal definition of B-1/B-2 status, we end at section 101(a)(15)(B) of the Immigration and Nationality Act, codified as 8 U.S.C. 1101(a)(15)(B):

an alien (other than one coming for the purpose of study or of performing skilled or unskilled labor or as a representative of foreign press, radio, film, or other foreign information media coming to engage in such vocation) having a residence in a foreign country which he has no intention of abandoning and who is visiting the United States temporarily for business or temporarily for pleasure;

This is not extremely enlightening, and unfortunately key terms such as "business" are not explicitly defined either in the act or in any regulations.

What we can notice, however, is that it is phrased in terms of the purpose of your visit, rather than as a restriction on what you're allowed to do. Your purpose cannot be to perform skilled labor, and you must be visiting for business or for pleasure.

If your purpose for going were something that's a recognized business activity, and the length and circumstances of your trip were commensurate with that purpose, then you would certainly be allowed to do things that are purely incidental to that purpose. It's not written down anywhere that a business visitor is allowed to eat and sleep in addition to performing his actual business purpose; neither is it written down that he can answer emails and keep in touch with his duties at home, but in this day and time all of that would be expected, natural activities of a business traveler.

You're on slightly less solid ground when it seems that you're proposing a longer pleasure visit, and want to keep doing that kind of incidental activities for your employer at home. I would still expect it to be okay, given that we're talking about activities where

• (a) you're not being directly paid by anyone to do them (since you say you will be on leave), and presumably you do them only in order to preserve your employability after you return,
• (b) you would be doing them at home anyway,
• (c) it doesn't benefit your employer that the things are being done from the US rather than at home,
• (d) you're spending such a small amount of your time on them that it doesn't cast doubt on your purpose for traveling being pleasure.

On the other hand, it won't do you much good what a random person on the internet expects if the authorities turn out to disagree with it. Eventually you'll need to either make your own decision, or possibly pay a small fortune to consult an immigration lawyer (who will not be able to give you any guarantee and may not even be able to tell you anything except "it depends").