I assume this is OK if done properly. There is already a dedicated circuit for the range and a box with a plate. The box location (i.e. where a plug would be) will leave a large gap behind the range given the configuration of the back panel of the range and protrusion of the plug. Are there special wire nuts that should be used?

Direct wiring of a range is allowed, but typically required to be installed in some type of conduit (ie flex) by code and the manufacturer. There should be some information about direct wiring in the manual. The conduit may cause some of the same interferences. Best bet may be to see if you can move the electrical box back or up. This could be easy or not possible depending on the current configuration.

A whole lot of "it depends" on the wirenuts. Assuming a 50A circuit, you probably need to look at a different connector than a wirenut - something like this: Link. There's a lot of consider, ie aluminum to copper connections, stranded vs solid, conductor size, etc. Someone else may have a better product, but most wirenuts aren't listed for a connection that big.

It's not generally good practices to have a splice in the middle of a run for a large direct connected appliance like this, so people avoid it. Some local codes may not allow it. If there's a different option you can consider, I'd recommend going that route.

Does your range already have a pigtail on it?

Hardwired ranges present the same shoulder/fat conundrum as a dryer vent. Gotta make sure you have enough slack in the cord to be able to get to it to make a connection - while accounting for big shoulders or fat guts - but not so much it all gets mushed up between the wall and appliance. Also want some slack to be able to move it to clean behind it or retrieve the jihadist salt shaker. Nice thing with plugs it you just need to be able to push or pull on the plug.

Been looking for a recessed range plug (but can't seem to find one that is suitable

Look for a 14-50R receptacle. Should be like $10 at Home Depot. If you have an older house, might need the 10-50R depending on the wiring. Need a two gang old work box to cut into the wall.