I think executive orders are a good indicator into the mind of a president and his administration. I wasn't able to see a sequential list, from newest to oldest. Where can I find this?
Wikipedia has a good list of executive orders here, which is kept reasonably up to date by the community. Further you can get to all of Trump's executive orders here, which will have references to all the specific executive orders as you need them.
As for live feed... Honestly I'm not sure what you are looking for there. I don't think there is any live feed service for executive orders except for maybe news alerts. Executive orders don't come out on a regular schedule, and they generally don't have any immediate (as in down to the minute) impact. So the need for a true live feed for executive orders isn't really there.
This is what I have:
However, it doesn't go into too much detail. Only covers the text of the exec order, doesn't appear to hyperlink to previous executive orders, nor any detail about authorship or background. Only shows the current President's (Trump) Exec Orders, not previous ones.
And there's no apparent way to live-feed this and get a text update SPECIFICALLY when a new one comes out.
Unfortunately, there is no requirement to publish executive orders as they are announced. The White House link is probably the closest to a live feed as you will find, but the media announcement (on TV) will often occur before the order is published on the website.
All executive orders will eventually be published in the Federal Register but that happens even later. Also, they aren't separated from everything else.
The Wikipedia link posted by @DavidGrinberg is probably the nicest presentation that you will find in terms of a readable list. But of course it is only maintained by volunteers, not by government officials.