Hmm, I actually really like this Demigod idea. I like the general ideas of most of the abilities, although I think they need tweaking, and I especially love the "Godnap" ability. However, I think that 20 yards for an attack range is overkill. Something similar to Regulus' would be fine.
Burrow sounds very interesting. I could see it being used to set up ambushes, and popping out to take advantage of the attack bonus. Instead of damage/second over time, though, I think a simple one-time damage bonus would work best. For example, at level 1, for each second burrowed the attack would increase by 50, up to a max of 400 or so, ramping up to 1000 at level 4. It should include a stun, too, after all it is a SURPRISE attack, right? On the other hand, you can't just let the player burrow at any time. If you do, nothing stops them from simply burrowing to escape death. A possible solution to that is to only allow a Burrow when no other Demigod's are around, and/or only allow it if the Usurper has not been attacked or been attacked in the last 10 seconds. *** (see below)
Improved Burrowing sounds a little unnecessary, even useless, to be honest. An ability like Burrow shouldn't have a very high cooldown to begin with, so reducing it would be futile. No one would buy this ability. I think it's best to scrap this one altogether and come up with a new ability.
Bask: Overall I think this is solid. I kind of like the trade-off between vulnerability/revealing your location and increased regeneration. It's hard to tell how well it would actually work in-game, though.
Godnap: My favorite. I can see this being used both to prevent enemies from escaping, and from literally snatching back-line support like QoT or Sedna from the rear of a battle. I don't think it needs to cost "significant" mana, a regular amount (500-ish would work fine.) I would make this one of his primary abilties, in other words an ability with more levels, available earlier on such as Level 2. I would also start the range at 15 and ramp it to 30, and add a stun to the attack. *** (see below)
Constrict: I envision Usurper wrapping himself around his enemy, snake-like, immobilizing them as he squeezes the life from them. I would increase the damage on this by a lot to make it more combat-useful since he doesn't have many direct combat skills. I'd make the time factor constant (3 seconds is good) and simply ramp up the damage per second each level, 100 damage per second per level.
Cold blooded: Sounds good, not sure I'd change anything.
Swiftness of the Serpent: Skill cooldown already exists in Sedna's aura, so I'd change this to be more distinctive. A perfect substitute would be a movement speed aura (the name still fits!), adding 7/10/15% speed. This fits better with his theme of swiftly hunting down enemy Demigods like the snake he is.
Venom: Another good one, no ideas to change it right now. Again, it fits with his anti-DG theme. All the Demigods rely on mana to some extent, and this would counter that.
***: A possible substitue for "Improved Burrowing" would be "Element of Surprise." A passive skill that adds a stun element to his Burrow and Godnap abilties. In other words, Burrow and Godnap would not stun anyone until you bought "Element of Surprise." Possible stat, 1 second of stun per level.
Also, one other suggestion: get rid of "Bask" and combine it with "Burrow." I can see Burrow being difficult to use, because it's pretty hard to predict where a Demigod is going to show up, except at flags, and no one wants to wait a long time for THAT to happen. Bask is pretty similar, because it requires you to sit around, doing nothing interesting, while it activates. If you give Burrow a regenerative quality similar to Bask (although not as powerful or else it'd be TOO good) you kill two birds with one stone. Plus, it kind of makes sense that he'd regenerate himself as he's chilling in his burrow. It also frees up a triggered ability slot that you could fill with another, more interesting and unique skill! (He needs an AOE, which he is notably lacking. Maybe something like "Tail Whip" where he lashes an area in front of him.)
Anyway, I sincerely like this Demigod idea. He strikes me as a great anti-Demigod character, kidnapping his enemies with Godnap, laying in wait for them with Burrow, ruining their mana with Venom, and chasing them down in packs with Swiftness of the Serpent. I think he's a solid character, but I really think you should take my suggestions to heart. I think they would do a lot to improve an already good idea, so consider them!
Edit: and finally, in order to make it fit into the Demigod universe more, you should gear the character design to be more humanoid rather than animalistic. For example, something like a Naga with the body of a snake but human body parts like arms...