Generally no. Most moving companies are scoped to transport items you're keeping, not to haul unwanted furniture to a dump or donation center. If you need that done, plan on a separate junk removal service or a municipal bulk pickup.
Most curbside recycling programs accept flattened cardboard, but many limit how much you can set out at once and require boxes to be broken down and free of tape. Check your local program's rules before stacking a large pile at the curb.
It's usually best to wait until you've fully unpacked and know exactly what you're keeping before booking junk removal. Scheduling too early often means a second trip once you find more items to get rid of.
Some do, especially for larger items like couches or dressers, but policies vary by organization and location. Call ahead to confirm pickup availability and whether your specific items meet their condition requirements.
Clean bubble wrap can sometimes be reused or dropped off at shipping stores that accept it, while plain packing paper is recyclable in many curbside programs. Foam peanuts typically aren't recyclable curbside, so check for a local drop-off option first.