I. The "skew" in this case refers to the moving map shifting your ownship position closer to the edge of the map to put more of "what's in front of you" on the screen. So, if your map was oriented north up, and you were heading north, your ownship would "skew" to the bottom of the screen so that more of the map "in front of you" would be visible. (Note that this behavior only occurs when you are moving, not when you are stationary.) "Center aircraft no skew" keeps your ownship fixed in the center of the map so that you can see an equal area of the map both in front of and behind your ownship.
This behavior is related to the "Auto Track Plane" setting (Menu > Setup > Map Options > Auto-Track Plane), which manages whether and after how long a delay iFly will automatically "snap back" to your ownship after you pan the map away from it. If you have "Center aircraft no skew" selected, your ownship will be centered on the moving map when it auto snaps back (or when you tap the "recenter" button on the moving map page, which appears as a crosshair with a green dot when you pan away from the auto-track/centered position). If you don't have that option checked, then when iFly returns to tracking your ownship, it will be skewed as I described to have your ownship nearer to an edge to present more of the map in the direction of your route of flight visible.
II. "Auto show airport diagrams" has to do with the behavior of the map as you zoom down into airports on the screen: One option is to always show the "RealView" (think Google Maps satellite view), but with "Auto show airport diagrams" selected, then if there's a Chart Supplement (A/FD) Airport Diagram available for an airport, that's what will be displayed at close-in zoom levels instead of the RealView image. (For airports with no FAA Airport Diagram, the RealView image continues to be shown when zoomed in.)
Auto Taxi is a different feature. When enabled, then when iFly detects that you have landed at an airport, either the Airport Diagram (if available) or the RealView (if no Airport Diagram exists) will automatically pop up at full-screen zoom level so that you can see your ownship as it is taxiing.
III. It sounds like you may be confusing the "Flight Plan Sharing" feature (a real-time direct device-to-device FP update capability, accessed (on the primary device) via Flight Plan > More Options > Share with Local Devices > Share Plan) with the "Sync Devices" data transfer feature (one-shot movement of chunks of data from one "source" device up to the iFly cloud in one operation, then down from the cloud to one or more "target" devices in a separate operation, accessed (on both the source and recipient devices) via Menu > Setup > Sync Devices...).
For Flight Plan sharing, when you initiate that on the primary device, you'll see a pop-up within a few seconds on the secondary devices to confirm acceptance of the shared FP. (This confirmation happens only when the real-time link is first established. By default, subsequent changes made on the primary device will get automatically and silently pushed to the secondary device(s), though you can change this behavior by unchecking the "Auto-Share Changes" option before clicking the final "Share Plan" button.) FP sharing does not involve the Internet--the devices only need to be on the same local network together, and data goes across that local network directly from one device to another.
For the other data upload/download function ("Sync devices"), the data transfers are not automatic or in real-time. Any time you upload new data to the cloud from one device that you want to be distributed to other devices, you have to take specific action on those recipient devices to go grab the data from the cloud. This obivously requires the device(s) involved to have internet access, since they have to reach out to the cloud (iFly servers).
Flight Plan sharing (real-time) is intended for devices on the same network that you want to stay in sync with the real-time flight plan over the course of a single flight (think a tablet on the pilot's yoke and in the copilot's lap).
Device syncing (data up/downloads via the iFly cloud) is intended to allow users to configure devices in the pre-flight environment with broader sets of user data (catalogs of stored flight plans, user-defined waypoints, etc.). Think a home-based laptop PC and a tablet device used in the plane, or getting all the data off an older device that's being retired and transferring it over to a newer replacement device.