I have started playing Ned's Island. Will I unlock all weapon slots and backpack slots without completing original content?
Your probably not going to unlock those if you stick to the zombi Island DLC. According to the borderlands wiki:
The Backpack SDUs each offer 3 additional slots of storage capacity. Each one is gained as a reward for completing one of the ten Claptrap Rescue Missions, for an extra thirty spaces of storage that can be earned throughout the game to a total capacity of 42 on the first playthrough.
Weapon Equip Slot SDU
Two Weapon Equip Slot SDUs are available as quest rewards in Playthrough 1. Together these increase the weapon equipping capacity of a character up to four weapons, allowing an improved selection of weapons in the thick of a fire fight. Each weapon equip slot provides self-contained 'storage' of one weapon in addition to the inventory capacity. The two missions that offer these as rewards are: Return To Zed Road Warriors: Bandit Apocalypse
Yes. The storage deck upgrades will be available to you regardless of whether or not you've played the original content.