The asgard core, while Sep may groan about it, is the tool to create advanced Asgard stuff. It contains all their knowledge, including the schematics and construction manuals for all their technology. They don't need to use earth computers, because the asgard computer that an asgard core is, is far, far better.

Some things can easily be built using the shipboard matter converter. Others would require more complex assembly systems. We would be very surprised if they do not have an Asgard core somewhere in Area 51 or a similarly secure facility that is on Earth. Given time, they would be able to slowly build up an asgard-level assembly facility to build most things. It might take them several years post-Unending to do so, but that they have had. It won't be as easy to construct such a facility underground, but not impossible. The more feasible alternatives to underground are offworld and in space. They can't very well set up an Asgard shipyard anywhere on the surface of the Earth. Anywhere outside Sol would be uncomfortably vulnerable to enemy attack, so that leaves underground or in hidden in space, probably in a Lagrangian point not visible from the Earth, or on the surface of another planetary body in Sol.

There's no reason the Tau'ri can't build Time-dilation Technology. Its more probable that they haven't. Or won't.
Its quite possible that the technology is fairly expensive power-wise. Ideally, a time-dilation field would be optimal for research or construction. or for training. Imagine someone going through years of training in a matter of days or weeks. Using that, they could train personnel rapidly. The real problem with this is that its too powerful. Like the Wormhole drive at the end of SGA.

Another example of too-powerful tools would be using a pool of replicator nanites as a factory. You could simply assemble a pool of them, then throw space rocks in and get refined raw materials out on the other side. Its obvious they understand a lot of the replicator code with how the IOA was able to modify it for the Ark of Truth incident.

The Stargate universe is rife with technology that can be (ab)used in overpowered ways by the right minds. For that matter, it would not be surprising if they've also gained access to certain amounts of Ori technology as a "thank you" for liberating (for lack of a better name) the children of the Ori.

But for the sake of having an enjoyable RP, most of these uses for potentially abusable technologies ought to be ignored.

[b]Sep said the core is complete, but with severely restricted access and some slowness (in part) due to the restrictions. Lets stick with that.[/b] Any appearance-wise differences between the various 304s are semantics. Other than the obvious ones like a ventral deployment platform on one ship.