if you are staying in a hotel or a resort and not going into a village it is easier. If you you are stopped, you might be asked where you are going but you will have documents from your ASQ. Should be ok.
As far as getting it back, depends on many factors such as any damage or cleaning fees. Also not all owners give it back. My experience is usually a 50% loss for made up reasons and standard cleaning fee.
You are required to formally identify yourself according to Thai Law. You are also required to show evidence of permission to stay in the Kingdom. So semantics about non specific immigration law wording is not going to get you far when asked by police for ID and valid permission to stay.
The law does not specifically say it. But you do need to produce it when asked by police. So i wouldn't be too far away from it. Like another province.