Here’s a list of things that can be used with the C-Gear:
The Xtransceiver can be used to communicate with other players nearby using wireless, where up to 4 players can talk to each other using a radio-like system and, if they have a Nintendo DSi, DSi XL, or 3DS and camera function is not disabled, see each other. The Xtransceiver can also be used over Wi-Fi by selecting it at the Global Terminal in Pokémon Centers.
The C-Gear can be used to warp the player to the Entralink, an area in the center of Unova. In order to access the Entralink, the player must be standing in an open area, a requirement similar to the use of Fly. Pokémon befriended in the Pokémon Dream World and items obtained in the Dream World can be obtained here. Players may also perform missions by entering another player's world, similar to The Underground. Players may also register Pass Powers to their Tag Log via the Entree in the Entralink.
The C-Gear can be used to synchronize the save file of the game card to the Pokémon Global Link site, which provides access to the Dream World, in a procedure called the Game Sync. The player will be asked to select a Pokémon from the Pokémon Storage System to put to sleep, which will then appear in the Global Link.
A Pokémon may only be put to sleep once a day (and can be awoken up to twice). Changing the DS clock or using the game in another DS will result in needing to wait 24 hours before a Pokémon can be put to sleep again.
These next four things are done with Infrared (IR).
There are four battle modes for two-player battles (Singles, Doubles, Triples and Rotation) and a Multi Battle mode for four-player battles. The rule set is fixed for all modes and they are as follows:
Rules for IR Battles
- Any and all Pokémon in the player's team will battle. (Event clause OFF)
- All Pokémon will become Level 50, regardless of initial level and their stats adjusted to their Level 50 ones. Pokémon do not evolve/devolve and will not gain new moves nor lose any. (50-All Rule)
- There may be more than one of the same Pokémon with the same National Pokédex number. (Species clause OFF)
- Two Pokémon on the same team may hold the same item. (Item clause OFF)
- Wonder Launcher is ON
Unlike Union Room battles, players cannot set the rules and players are unable to preview the opponent's team.
Trading is now done with the whole of the player's PC box, and players can choose up to three Pokémon for the other party to choose before settling on a trade. The boxes can be selected by a slider at the bottom, and there are emoticons at the bottom left-side for the player to use while selecting Pokémon. Pokémon which know HMs cannot be traded out of a player's party.
In a Feeling Check, players are shown how compatible they are by performing the Rhythm Check and the Sync Check. Players are required to tap 10 times at the touch screen for the Rhythm Check, and press both their thumbs against the touch screen for the Sync Check. At the end, the player's compatibility is given a score and a ranking. Both players get Sweet Hearts which can be collected and exchanged for a Heart Scale at Mistralton City (10 Sweet Hearts for 1 Heart Scale); in Black 2 and White 2, on Valentine's Day (February 14) and White Day (March 14) the player can exchange 5 Sweet Hearts for 1 Heart Scale.
Players are also able to view the top 30 scores of their Feeling Checks at the "Ranking" section. (For more info, click my source and scroll to “Infrared”)
After both players have obtained their Friend Codes over Nintendo Wi-Fi Connection, they may register each other's friend code in the Pal Pad through IR. Up to 32 Friend Codes may be registered.
At the bottom(BW)/center(B2W2) of the C-Gear is a bar called the Tag Log. The Tag Log keeps tracks of the people the player has passed by during the time the C-Gear is on, provided the other player has their C-Gear on as well. For every person that the player passes by, a colored marking will light up. Every time a different player passes by, the colored marking will shift to the right, and only up to 10 colored markings will be shown.
When the Tag Log is selected, the details of the people that passed by will be shown in small boxes: their in-game name, the Trainer class the player has selected on their Trainer Card, and up to 5 orange dots that correspond to the player's statistician ranking. (click my source and scroll to “Tag Log”)
When the small boxes are selected, they will show more details about a passerby, which includes his or her in-game time and current location, occupation, favorite activity, number of people that player has passed by and thanked, and a customized quote. The information shown will only be updated to that point in time when the players have passed each other, and will not change even if either player has changed it at the Passerby Analytics HQ. The passersby' name will gray out when he or she is inactive or is not within wireless range of the player, but the information will still remain in the C-Gear.
The bottom of the Tag Log also shows the total number of passersby the player has encountered ("Total:"), a downward-pointing triangle which activates the player's stored Pass Power from the Entralink, a speech bubble which sends the player's customized message to all of the passersby in the Tag Log, and a smiley face which sends the player's "Thanks" to a selected passerby.
It may be interesting to note that when certain conditions are met, specific Pass Powers will activate: (again, click my source, you should know where to go next for this section)
In the bottom-left corner of the C-Gear is a pie chart icon which opens the Survey Radar. In the Survey Radar, the player may select the active survey, which they will receive answers to from players passing by.
The player may also analyze the previously collected results of the currently selected survey. By tapping different text boxes on the touch screen, the player may switch between the questions in the survey, to analyze them all; switch between answers for the currently analyzed question, to see how many and what percentage of people gave that answer; and switch between viewing only answers from the people the player has met that day and viewing all the answers the player has collected for that question.
These next two are for the “Customization” section.
The C-Gear may be customized in a number of ways. By touching the C-Gear logo on the top of the screen, the style of the icons can be changed. The arrangement of the icons may also be changed by touching the wrench icon, then spaces on the screen to change the function of the icon. The colors of the rings and the C-Gear may also be changed by touching the wrench icon, then tapping the rings or the C-Gear.
By default, players start out with the Basic Skin for the C-Gear; however, additional C-Gear skins can be downloaded from the Global Link to further customize the C-Gear. Skins mainly feature a certain Pokémon. There are certain ones that don't necessarily feature a specific one though, such as the Pokémon Café skin. With the closure of the Generation V Pokémon Global Link on January 14, 2014, players are no longer able to change their C-Gear skin.
Bulbapedia says BW/B2W2 are 100% compatible, except for the Entralink, because it has no service, now. They didn’t say anything about the Battle Subway.
To answer one of your questions:
You won't be able to do the Entralink missions from Black/White versions.
Black 2 and White 2 have different missions.
However there are plenty of other options.
Bulbapedia doesn’t give in depth details of what “connect” means but:
Black and White are able to connect with all fourth generation core series games, as well as their sequels Black 2 and White 2.