I found a bug in dhtmlXCombo when using z.enableFilteringMode(true);
The pull-down arrow image is not show. This is both in my code and your provided samples.
When removing z.enableFilteringMode(true); the image pops up again.
Is there a way to fix this? 
Actually latest version shows drop down image in any mode
dhtmlx.com/docs/products/dhtmlxC … ilter.html
In initial version of dhtmlxCombo, image was hidden in case of filtering, you can comment next line in code ( or just update to latest codebase )
this.DOMelem_button.style.display=(this._filter?“none”:""); // line 125 in dhtmlxcombo.js