1. Yes. That is the bad. You should eventually replace the BNCs to the isolated ones.
2. OK. I like to emphasize again that everyone works on electronics should read data sheets more carefully and seriously because they have many important practical instructions to exploit full performance of the components.
1. Why do all of the BNCs have no GND connection? Each should have the individual cables to the ground. Each signal line and the corresponding ground line should be twisted together.
2. This looks the (usual) oscillation of the V-I conversion stage but I can't tell anything as I don't have the circuit diagram of the whole circuit.
3. In a certain case, putting some capacitance at the feedback may help. Read P.11 of the data sheet of LT1125. Try to put some capacitors from 20pF to some larger one whether it changes the situation or not. I suppose the bandwidth of your sensor can be ~1kHz. So putting a capacitance less than 10nF still has no effect to the servo.
1. They are all connected to the box which has a single connection to the board ground. If I connect each of them to the ground, there would be many small loops
of ground. Of course, I should have replaced all the connectors such that the they are disconnected to the box as point out by Robert.
2. The oscillation disappears after I add 5 nF capacitor in parallel to the existing resistor. Thank you very much for pointing this out.