Using e-tube software or a cockpit junction (SM-EW90-A or –B), the Ultegra rear derailleur can be adjusted += 16 clicks.  I estimate that one click moves the chain guide pulley about 0.125 mm.  Total range is +/- 1 mm.  The equivalent adjustment on a mechanical system is turning the barrel at either the derailleur or the shifter.

Yes!  You can program Alfine or Ultegra to shift twice or thrice when a button is pressed and held.  You can also control how fast the multi-shifts occur.  The fastest setting is very fast indeed.  I don’t use multi-shift, which means when I press and hold the button, the derailleur performs a sequence of single-shifts until I release the button.  I also configure it for the fastest setting, which  works fast and well for my Ultegra derailleur.  I can shift a bit faster yet by rapidly pressing and releasing the button instead of holding it.

I could never tell from the pictures I saw whether this was one button that moved two ways or two buttons.  It is actually two buttons.  By default the big, obvious button shifts up.  The plastic around the button is hinged on the left—you press it on the right side to shift down.  The E-Tube Project software can be used to, among other things, reverse the action of the buttons.

Whenever the display is active it helpfully shows the gear you’re in and the state of charge of the battery.  It also stares back at you with a useless “M”.   You won’t find this in any of the Alfine documentation but I’m pretty certain the M stands for Manual.  As opposed to S for Synchronized.  The new (as of May 2014) XTR Di2 system for mountain bikes introduced Synchronized Shifting.  With a Di2 front and rear derailleur, synchronized shifting controls both with a single shifter.  When, for example, the smallest cog is selected in both front and rear, the next button press for a shift up will simultaneously shift the front derailleur up to the next biggest cog while shifting the rear down enough cogs to achieve the next incremental increase in gear inches.  Pretty slick but not applicable to a system with rear-only shifting.  It would be nice if the software let you turn this indicator off but I haven’t found a way to do that.  If you’re running an XTR Di2 system with synchronized shifting you can switch between manual and synchronized shift modes on the fly, in the latter mode I’m pretty sure the indicator will switch from M to S.  I’m also pretty sure the indicator will show which of two customized shift mappings are selected when in synchronized shift mode.  Note that while I’m partial to internally geared hubs, the XTR Di2 system should work great on a trike.  The original road Di2 systems never had enough gear range for a recumbent but I’m sure the XTR system does.

I’m guessing you can but these components represent a hybrid of what Shimano considers MTB and Comfort so it’s hard to say whether the software will support the combination without having all the parts in hand.  I intend to answer this question at some point, and whether synchronized shift is supported.

Possibly not.  In the course of ordering all of my Di2 parts I wound up with the plug tool and two plugs.  I believe the tool came with the shifter switch and a plug came with each of the motor unit and battery but I could have these mixed up a little.  I ordered a tool before realizing this so I have two.  The plugs are essential for any unused e-tube ports, to keep them waterproof.  The tool is handy and works well but you could get by without it if you’re careful to keep the plug perfectly aligned when plugging or un-plugging.  That said, I use the tool for plugging because it reduces some risk of mucking things up.  I usually un-plug without the tool but in tight clearances on the bike the tool might be essential.

No and Yes.  When Shimano introduced the internal battery for Ultegra and Alfine, they graciously added a USB charging port that also connects the Shimano software to the system.  This is the same software that drives the programmer and all of the same things can now be accomplished with the charger instead—changing shift behavior, updating firmware.  Except one:  checking the system for errors.  If you want to check the system for errors, you need the programmer connected to an E-Tube port in the system.  I probably won’t bother with the programmer.

Note that to use the charger for configuring the system and updating firmware, you need either the display unit or SM-EW90 cockpit controller (A or B) in the system because these are the pieces that have the charger port.  If you have an external battery you can still use the charger to program the system but it won’t charge the battery.  I don’t know if it will program the external battery.

Yes.  The SM-JC40 and 41 junctions are “dumb” electrical/mechanical connections that can be daisy-chained.  These are very good connectors but even the best compromise overall system reliability to some degree.  Use them sparingly.

This gets complicated.  What I currently know is that the current generation of Di2 shifters will work with any of the groups.  You cannot, however, mix front and rear derailleurs between groups (Ultegra rear derailleur with XTR front for example).  Older generation(s) of Di2 used different connectors and bus architectures and are not compatible with the current Ultegra, XTR, and Alfine E-Tube components.

It holds the sprocket to the hub.

Nothing.  It stays in whatever gear it was in.  When you plug it back in it remains in the same gear.

Here’s the block diagram of the configuration I initially tested.  I’ve since replaced the Alfine motor with an Ultegra derailleur.  You can also replace the junction and display with a single Cockpit Junction Box (SM-EW90-A or -B). Though these alternative configurations are a mix between the Comfort and Pavement groups, they are all compatible.

It’s a spacer for the Centerlock disc brake.

Based on the E-Tube software I’m confident you can combine a primary shifter with one or more satellite / sprint shifters within the Comfort group that is used to configure the Alfine components.  I haven’t tried it though.

No.  The SM-EW90-A or –B works great as a replacement for the display.  Your shifter plugs into it instead.  The E-Tube software recognizes it as part of the Comfort group.  By flashing and changing the color of a single LED, it shows the same battery life information as the display.  It has a charging/communication port just like the display.  It just doesn’t indicate the gear you’re in.  But you cannot have both this and the display connected at the same time—shifting won’t work and the software detects the configuration, telling you it is not allowed.  Note that Shimano now sells the SM-EWW01 Di2 Wireless data transmitter which will send (via ANT+) gear position and battery state of charge to compatible cycle computers like the Garmin Edge 1000 and Scio Ant+.

Yes.  The E-tube software implies you can have as many as 6 satellite/sprint shifters and 6 “primary” shifters like the SW-S705, all within the Comfort group.  I haven’t tried more than one though.