Yes, standing a mattress upright and strapping it securely against a wall inside the moving truck is one of the safest transport positions. It avoids the uneven pressure that comes from stacking boxes or furniture on top of a mattress laid flat, which can create permanent dents or sagging.
A mattress bag itself can stay on indefinitely since it only wraps the surface, but if you are using a compression-rolled foam mattress (the kind shipped rolled in a box from the factory), it should be unrolled and allowed to expand within a few days. Extended compression beyond what the manufacturer intended can affect the foam's ability to fully decompress.
For most local moves, a fitted plastic mattress bag is sufficient protection; a full box is unnecessary unless you are placing the mattress into long-term storage where extra rigidity helps prevent bending from stacked items nearby.
It is not recommended. Roof transport exposes the mattress to wind pressure, UV fading, and moisture, and straps alone often cannot prevent shifting at highway speeds. If a truck or van is not available, it is safer to make two trips or arrange for a moving crew with proper cargo space.
Vacuum both sides and the seams with an upholstery attachment to remove dust and allergens, spot-treat any visible stains with a mild fabric cleaner, and let it dry fully before sealing it in a mattress bag. Moving a damp mattress into a sealed bag encourages mold growth during transit.