NFA-epsilon: epsilon loops, and how do I deal with them?
if you had a chain of epsilon-arches, would you just jump to the end of the chain instantaneously


I am not sure for "instantaneously", but I would say it accepts the empty string $\epsilon$.


  would this cause a loop and in-turn cause the string to never halt, or would it just create another fork (endlessly?)?


Your input string will be accepted. On any input, a NFA runs by creating many branches of computation in parallel and if one of them reaches the accept states, after it reads the last symbol of the input, then your string is accepted and the entire computation halts. You kind of choose at least onе fork that leads the NFA to the accept state. 


  how would I approach said NFA-epsilon example with these example strings: 1011100, 0101111?


Right the first string is not accepted, the second one is accepted.