Can I put BBQ ash in the HOTBIN?

Can I Add BBQ Ash to my Compost? 

DO NOT add hot ash to the HOTBIN, or any other compost bin - it will damage the plastic and void the guarantee. It can ignite a fire if your heap is very dry.

Can I add cold BBQ ash to my compost?

Yes, but too much ash will turn the compost waste strongly alkaline and this will slow down the composting process. Ash (and especially BBQ ash if lots of fat has dripped in towards the end of the BBQ) can also form an impervious layer if not mixed in or too much is added in one go.
 
How beneficial is BBQ ash to the heap versus directly adding it to the soil?  

BBQ ash won't 'breakdown' (ie compost), but it will be incorporated into your compost. BBQ-ash is a mixture of minerals including potash (ie potassium oxide), the source of the second item in NKP fertiliser list. Potash (like any fertiliser) has to be used sparingly. Spread 'thickly' around stems/roots it will 'burn' as it is highly alkaline - so there are some advantages of 'diluting' the potash into your compost.

Can I add BBQ briquettes (broken bits, leftover dust, etc) to my compost?

No. As you add waste in the HOTBIN it can be come heavy and its this weight and pressure that could cause the briquettes to become crushed (and dusty) causing problems with aeration as airflow becomes blocked in the unit. Further chemical accelerants are often added to the briquettes so it is also a case of not knowing exactly what is being added into the bin.

Charcoal is a natural, non-toxic, inert form of carbon. It is used widely in agriculture (especially in Japan), is believe to be one of the critical components in Amazonian Terra Petra (Dark Earth, or ATP) and is closely related to Biochar. A huge group of environmental scientists are saying actively add charcoal to soil!