Using plans to build a bed headboard which calls for 3/4" hardwood plywood (34" x 60 1/2"), which is quite pricey for a sheet. What would the main purpose of hardwood specifically be, besides being more solid? I'm wondering if I can get away with pine plywood 1/2". Will that bend too much or something? I was wondering if I could even get away with 1/4" with some horizontal 2x2's for support behind it. The cheapest option may be to instead use 2x6's for the entire headboard but I'm wondering if that will be way too heavy.
You can certainly use pine/fir plywood. It depends entirely on the appearance you're after. Pine tends to look more casual or rustic than fine hardwoods. It's also softer on the surface veneer, which is why the best furniture tends to be mahogany, oak, maple, cherry, etc.
It's not substantially less rigid, though. Hardwood plywood is usually only hardwood on the thin outer veneer. The core of a BC grade pine and hardwood plywood is usually very similar. 3/4" is plenty strong and stiff for your application. There are high-grade plywoods with very thin layers, which makes them more stiff and stable, but that's a different situation than just switching species.
Whether you can reduce to 1/2" depends on the design. It'll be less flat initially and more likely to warp over time if not supported by a rigid frame structure.
The plywood plant I worked in made hardwood faced plywood. The core section(s) were fir. The reason hardwood is suggested for furniture is for looks. There may be all hardwood types out there but I know several plants in the Pacific Northwest only face the panel and some times on only 1 side.. as for the thickness that is where your strength comes from the thicker the more plys so you do not want to go thinner. You could build it from regular sanded plywood and purchase a sheet of veneer but that would end up costing more than pre made.