1. Is there a way for when executing a cue, for it to update automatically and bring up that cue when in "cue tab"?
In general, the "Live" display shows what is happening on stage and the "Cue" display is a blind editing mode. You can view any cue in the editing display regardless of the cue being sent to the output. To go to a random cue in Cue mode use the command line, press "q" which expands to "Cue:", enter the cue number, press "enter". Also, clicking on a cue in the cue sheet will show the cue in the "Cue" display mode. But, it also sets that cue as the one to be executed next by the Live "Go" button.
2. Is there a way to record what you have brought up in the "live tab" into the cue you are currently sitting in?
Yes, there are two command line functions that record cues. They work in both cue and live modes. The "r" key expands to "Record:" and then you enter the cue number of the cue that you wish to store the current levels into. In Live mode, pressing "r" then "enter" will always record into the current live cue. You get a warning whenever the cue you are recording into already exists. The other command is "c" which expands to "CopyTo:" This command is like record except that it also copies time and other information along with the levels into the destination cue. For example, suppose cue 4 is a 0 second blackout with "B/O!" in the cue's what field. You are in cue 4 either live or cue mode. Pressing r <Record:> 10 causes cue 10 to be black but cue 10 will have the default time and nothing in its what field (unless it already existed then the time and what field remain what they were before. In contrast, pressing c <CopyTo:> 10 causes cue 10 to become a duplicate of cue 4 with 0 time and "B/O!" in the what field.
3. Is there a way to have the channels you bring up in the "live tab" also be brought up in the "cue tab" so that you may record them into a cue?
Yes, as in the answer to the question above. There is a difference in editing in Live vs. Cue. That is that editing in Cue mode is direct and changes are made to the cue without the need to record them. In Live mode, changes are reflected in the output but require you to record them into a cue for you to save them.
4. Is there a way to be in the "cue tab" but have those channels you bring up when building a cue, also be brought up live, so when cueing a show you see what you are building?
No, that is what the live display is for. The general idea is that you build a cue in Live and record it. If in the process of playing back the cue you modify it, then you can choose to record your changes to save them. But, you may want to look at or modify a cue without having that cue appear onstage and that is what the "Cue" display is used for. You might think of the Cue display mode as being equivalent to "Blind" mode on other consoles.
The online help has a complete command line reference. Click on "Main Window" and then from the sidebar choose "Command Line".