You could put a strain gage on each leg. Used in a Wheatstone bridge configuration, it outputs a voltage proportional to the force in the leg. This could accomplish everything you want, and assuming you have relatively low accuracy requirements, the circuit is an op-amp and some resistors to give an analog input to a microcontroller or ADC.
18 servos is a lot of power no matter their size but you are going to need to be more specific with what kind of servos you are talking about. That said, this What is the best way to power a large number (27) servos at 5 V? is a good answer to your question.
1.) Is there material out there that explains how the frame can be translated from the center of the body to the legs?
When you say I've deduced this DH table into its corresponding matrices - presumably these matrices you mention are homogenous transformation matrices that capture the four movements implied by the DH parameters ($a, \alpha, d, \theta$). ...
Yes, there are servo controllers capable of even 24 servos. As example, check out Pololu Mini Maestro board. It communicates with RPi through USB or UART and You can find more information about that in this blog or Pololu Maestro user's guide.
These servos, as you said, are rated for max. 7.2V input voltage. 2S LiPo battery has a nominal voltage of 7.4V and ...
Hexapod robot with tripod gait cycle will be both more stable and faster in terms of the locomotion speed. It is more stable because the support polygon of the tripod gait is a triangle. But in return you have 6 more joints and hence 6 more servo motors, which increases the power consumption and the complexity of the system.
There might be a number of problems:
The power might not be enough. Perhaps the batteries are drained.
The servos might not be strong enough.
The servos might not be in positions to give them enough leverage.
EDIT: Upon rereading your question, it seems obvious that your problem is due to a lack of power. You cannot supply enough power from 4 AA cells in ...
It does not matter whether the legs are mounted in a circle or not. You should be using inverse kinematics to place the tip of each foot on the ground.
If you know the dimensions of your robot and share the hexagon arm is attached you can account for the joint offset in your calculations. After that, there is some geometry involved to solve for the angles....