The general advice we give is that you should always insulate between and above the rafters (warm roof) or between and under the rafters (cold roof).

Building Regulations
From Part L of the Building Regulations (England and Wales) or Section 6 (Scotland), we recommend that the U-value of a pitched roof, insulated at rafter level should be 0.18 W/m2.K for a domestic refurbishment. For a new build this should be 0.11 W/m2.K in England and Wales and 0.15 W/m2.K in Scotland.

If a ventilated roof is being constructed, there is a requirement to have a 50 mm fully ventilated air gap between the insulation and the sarking felt, in order to avoid condensation. If you are constructing an unventilated roof without the use of counter battens, space would need to be left for the breathable membrane to drape, both of which would impinge on the space available for the insulation.

Rafter Depth
Common rafter depths range from 100mm to 200mm. This limits the maximum thickness of the insulation between the rafters to between 50mm and 175mm. This generally would not be a sufficient thickness to achieve the U-values required to meet Building Regulations and more insulation would need to be added.

Adding another layer of insulation, either above or below the rafters gives another key advantage – it reduces the effect of cold bridging through the timber rafters. Obviously heat will be lost through the rafters at a faster rate than it is lost through the insulation. The amount of heat loss through cold bridging can be minimised by using another layer of insulation either above or below them.

So if you are insulating a pitched roof, we would recommend having insulation either above or below the rafters.

An example would be if you have 150mm rafters you would require 100mm between and 50mm below to meet required U values. All joints should be taped and the insulation should fit tight between the rafter.

In short, no there shouldn’t be any gaps between the boards or between the boards and rafters.
The boards should fit snugly between the rafters. They certainly shouldn’t be supported by nails as you describe. All joints of the boards should be taped and sealed.
There should be sufficient ventilation space above the boards for airflow.
I would call whomever fitted them and request that they rectify their mistakes. If they don’t answer your requests, report them to trading standards as it does not appear that you have received an acceptable standard of work.

The answer that was given is exactly the answer I would have given good answer