Do Left handed bikes feel different to ride? Want to buy a BMX bike but they only have left hand drive ones in stock. Does it feel the same as a right hand drive?
Pretty much yes. Depending on quality and material of the cranks you might feel the slightest difference when engaging because the stiffness of left crank+sprocket isn't exactly the same as just right crank. But practically: yes, the same. And, pretty much the main reason left hand drive exists on bmx in the first place: with the advantage that if your prefferred grinding side is right there's less chance of ruining the sprocket and chain.

With regard to pricing/options: the vast majority of BMX brands today sell both RHD/LHD parts and do so for the same price. At least I checked the major brands on a couple of websites and found none which had different pricing and none which didn't offer both options.

Hubs usually are available as either LHD or RHD for the same price; in the production process there isn't anything intrinsically different apart from e.g. thread direction, and LHD doesn't seem uncommon enough to be more expensive due to lower demand. The so-called flip-flop hubs work both for LHD and RHD and are more expensive due to needing more manufacturing/material. Hubguards these days are non-threaded so the exact same for left and right, idem for sprockets and chains.

Most of the cranks sold today are symmetric with the threaded slot for the sprocket bolt available on both cranks. Which also makes sense because that means apart from the different threading for the pedals the production process for both cranks is the same.