This should be fixed with:

http://hg.netbeans.org/main/rev/747f13bffadb

Integrated into 'main-golden', available in NB_Trunk_Production #341 build
Changeset: http://hg.netbeans.org/main/rev/747f13bffadb
User: Matthias Schmidt <schmidtm@netbeans.org>
Log: Fix for #141176: CC not offering constructors

Ok, now I see constructors in completion, but after I press Enter on some of the items, nothing happens.

I've checked in this:

http://hg.netbeans.org/main/rev/67489050bba8

for correct parameter completion for constructors. As a matter of course, plain methods will get this as well in the long-run.

But this won't pre-set the parameters to names, since no codetemplate variables work in groovy - neither programatically via getCustomInsertTemplate() 
nor do they work in the (groovy-)editor in general.

The code-templates have been introduced with: http://hg.netbeans.org/main/rev/86b4c99c2638

Probably, there's something crucial missing to get them work right. I'm kinda at my wits' end here ...

Tor, can you comment?

Your CodeCompleter method doesn't implement the resolveTemplateVariable() method:

org.netbeans.modules.groovy.editor.completion.CodeCompleter:
    public String resolveTemplateVariable(String variable, CompilationInfo info, int caretOffset, String name, Map
parameters) {
        return "";
    }

(By the way, try returning null instead of "" if you don't want to support any of your own variables but simply want the
existing values left alone)

Ok, this one is fixed with my changes and Petr's fix to resolveTemplateVariable() :

http://hg.netbeans.org/main/rev/1e509344f82c

there are other issues but I'll file them separately, so this one is v.