Ad 1: Typically one BTS can handle several cells. Common patterns are a one BTS covering a circular area with one round-radiating antenna or a three-sector BTS which covers three cells with sector-radiating antennas. One cell can only be handed by one BTS at a time. Two or more BTSes are not possible since the radio communication would interfere with each other. Note that this is completely different in WCDMA/UMTS since there is no concept of cells.
Ad 2: Since one cell is covered by exactly one BTS, the cell id uniquely identified the concrete BTS.
Ad 3: Since the BTS does not contain any control logic, the mobile communicates directly with the BSC, e.g. about radio resources.
Edit after comment:
1/ The BTS is "dumb" to say it simply. It does only what the BSC instructs it to do. E.g. The BSC tells the BTS as well as the mobile which frequencies to use for the radio communication. A BTS does not route traffic as it is hooked to exactly one BSC. It even does not route traffic to one of several mobiles attached to the BTS as this is done by the BSC. Think of the BTS as a Um-to-Abis physical layer and protocol transcoder.
2/ Actually my earlier statement that UMTS has no cell concept is not exactly true, it's just different.
GSM is FTDMA (frequency and time division multiple access). The radio channel is shared by using different frequecies (per cell) and timeslots (per mobile). Since radio frequency is used to distinguish participants, great care must be taken that not two GSM participants use the same frequency at the same time at the same location. The solution to this is cells, where geographic areas have different frequencies assigned. Network planning must ensure that no two neighbouring cells use the same frequencies as this may lead to interference since you cannot control exactly the size of a cell (e.g. due to absorption and reflection). In GSM, a BTS has a fixed number of radio transmission channels, the number depends on the BTS hardware configuration. If all channels are in use, the cell is full, this is indpendent of the location of a mobile in the cell.
UMTS is CDMA (code division multiple access). The radio channel is shared by encoding the payload in a way that allows to decode it later even if several senders use the same frequency range. That requires coding schemes which are collision free (all codes are different from each other to avoid senders using too similar codes) and a great deal of signal processing. As an analogy: on a party you can understand someone accross the room, even if ten people are talking. The more senders communicate within the cell, the smaller the cell gets in order to allow the BTS/Node-B distinguishing between senders. Therefore, in UMTS a cell size is not geographically fixed. The cell "breathes" depending on its load.