1. They can work full duplex but you have to manually configure them so one uses the low frequency pair and the other uses the high frequency pair. At 300 Bauds it is possible to fit TX and RX pairs 'one above the other' within telephone bandwidth. There is no automatic configuration method that ensures they work properly and obviously if they are not set to opposite band their signals collide.
2. Yes, but note that acoustic modems basically became redundant around the late 1970s. Having used them I can confirm they were horrible, even someone sneezing nearby could break the data link!
3. They can talk directly to each other if correctly configured. A server isn't required. Where it becomes difficult is the telephone line itself, acoustic modems use the telephone handset microphone and earphone so the 'hybrid' in the phone electronics is responsible for splitting the transmit and receive paths. If you just join modems together with a cable, they won't work.
As there is only one pair of wires carrying DC, ringing voltage, transmit audio and receive audio to a telephone they have a method (using transformers and resistors in old style phones) to route incoming audio to the earphone and microphone signals outgoing to the line. Basically the line contains a mix of both but as you know what you are sending, if you subtract it from the line signal, anything left must come from the other end.
Brian.