Yes, and most of the village is too young for heritage rules to object. No listed building here is above Grade II. There is one conservation area, and it covers Turners Hill, not the post-war closes. Hundreds of planning applications go in every year. The limit on your extension is more likely to be your budget than the heritage rules.
There are 15 listed buildings in the area, and all 15 are Grade II. There are no Grade I or Grade II* buildings. Grade II is the most common listing and covers buildings of special interest. It is still a legal constraint: altering a listed building inside or out, in any way that affects its character, needs listed building consent on top of any planning permission. Only fewer than one in ten homes predate 1930, so the listings cover a small group of older properties. If yours isn't one of the 15, listing won't affect your plans. If it is, even the windows need permission.
There is one conservation area locally, at Turners Hill, which this page also covers. Inside a conservation area, changes that would be routine elsewhere can need consent, because the council is protecting the look of the place as a whole: some demolition, some work visible from the street, and work to trees. Councils can also remove particular permitted development rights there. Check the Turners Hill boundary and any extra rules with the council before sketching a front extension. Outside it, and that covers most of Crawley Down, those extra controls don't apply.
The local planning register shows how much building work the neighbours are already doing. It lists 1,298 applications between April 2024 and June 2026, and 579 of them in the last twelve months. That is not the paperwork of a place where nobody dares touch anything. It is a village where the loft conversion and the side extension are standard household projects. The register doesn't say how those applications ended, so the volume tells you people try, not how many succeed.
In practice, for the typical post-war semi or detached house, standard permitted development rights cover a lot of modest work without a full planning application, such as some rear extensions and loft conversions within set size limits. Anything larger, or on a listed building, or inside the Turners Hill conservation area, needs the proper route. Confirm what applies to the exact address with Mid Sussex's planning team before you commission drawings. Being told no after you've paid an architect is an expensive way to find out.
Crawley Down gives you plenty of freedom to extend. Heritage controls apply to a small number of buildings and one corner of the area, and the rest can build much like anywhere else, within the usual limits.