Holant problems and holographic reduction: simple graphs or multigraphs?
My concern is about the confusion between the use of the word "graph" to mean either a) a simple graph, without self-loops and parallel edges or b) a multigraph, that can have self-loops and parallel edges (i.e., multiple edges between the same pair of vertices).


In the literature about Holant problems, the word "graph" should be read as "multigraph" unless otherwise stated.  As you noticed, I tried to be more explicit about this in my thesis.  In my experience, different areas of math use terms mostly consistently within their own area but in ways that would be contradictory when considering multiple areas and read strictly.

In your first example, the proper notation is $\mathrm{Holant}([x,y,z]|[t_1,t_2,t_3,t_4])$.  This means that the input graphs are bipartite with one part having only vertices of degree 2 (and assigned the symmetric binary function $[x,y,z]$) and the other part having only vertices of degree 3 (and assigned the symmetric ternary function [t_1, t_2, t_3, t_4]).

Also (and I am sure you know this, but to be clear), a bipartite graph cannot have self loops.  So if a bipartite graph is not a simple graph, then it is because it has parallel edges.

1) Yes

2) Not immediately.  I think you have correctly observed that their reductions work for simple graphs.  It remains to verify if the hard problems that they were reduced from are also hard when further restricted to simple graphs.

3) There is no general fix.  In general, you have to reduce from problems that are hard for simple graphs.

4) Gadget 2 does not satisfy your condition.  Consider $N_1$.  The only two vertices of degree 2 without dangling edges have the same neighbor set.