Well you could attach your hide/show functions within the success parameter of the ajax method.
var showMethod = function () {
// show method code...
}
var hideMethod = function () {
// hide method code...
}
$.ajax({
type : 'POST',
data : request,
success : function(response) {
if (response.yourvalue == false) {
showMethod();
} else {
hideMethod();
}
}
});
Since it looks like other fields need also to be affected if your main field will be visible or not, you can simply apply the same for all these fields at once, and don't wait to first switch the visibility of your main field to decide if the others should display.
Update:
I am bit confused now with what you want... You can't hide something in the xml, only attach certain parameters, that will translate to certain features for each field output. The showon will make a field visible when you have a certain value on another field. So the fields will be hidden till this occur. This is js and css. You can also use the Labelclass to attach specific classes to your fields https://docs.joomla.org/Form_field#Common_attributes. Generally, behind showing/hiding elements is css anyway.
On the other hand, I haven't try this but maybe you could simply set the showon parameter in those fields that you want to display, based on the value of your "ajax field" and see if those will appear when you get false.