How do cycles of non-consuming transitions in NFAs influence their halting properties?
No, the $\lambda$-loops you talk about are not forbidden.  But they do mean one has to be careful.

The actual way to determine if an NFA accepts its input is as follows.  Note that this takes linear time in the length of the input, not constant time as you say in the question.  First, for each state&nbsp;$q$ of the automaton, use a graph reachability algorithm to compute the set of states&nbsp;$\Lambda(q)$ that can be reached from&nbsp;$q$ using only $\lambda$-transitions. For a fixed automaton, with a fixed number of states and fixed transition function, this can be done in constant time. Note that $q\in\Lambda(q)$ for every&nbsp;$q$, since you can reach&nbsp;$q$ from itself by doing some number of $\lambda$-transitions: specifically, that number is zero.  This means that $\lambda$-transitions from a state to itself are completely harmless.

To determine whether the automaton accepts its input, we just keep track of the set of states that the automaton "might" be in after reading each successive character of the input.  Initially, it could be in any state in $\Lambda(q_0)$.  If, after reading some characters, it could be in any state in $S\subseteq Q$ then, when it reads character&nbsp;$a$, it could be in any state in $\Lambda\Big(\bigcup_{s\in Q}\delta(s,a)\Big)$. It accepts if, at the end of the computation, the set of states it might be in includes at least one accepting state.  Note that, at any time, the set of states the automaton could be in is bounded by the fixed constant&nbsp;$|Q|$ so all the manipulations needed on the set of possible states can be done in constant time per character the machine reads, independent of the length of the input.