Widget

If an effect such as Takahide's Spinning Blade makes spaces obstacles for enemy units, can Widget still move Pyro into those spaces?

No. Widget, a unit, is performing the action, so for Widget that space has an obstacle, and two obstacles can't share a space (page 18).

Is Pyro immune to ranged attacks, since he's a token?

Not immune: immunity is a specific keyword. But attacks can only target enemy units (heroes or minions), and Pyro is neither (page 13). See also the Clear action on the same page.

On Widget's cards, is range counted from Widget or from Pyro?

From Widget, who is performing the action, unless the card says otherwise. Cards that count from Pyro say so, e.g. the Breath greens: "in range of Pyro".

Does Pyro continue to live if Widget dies?

Only tokens and markers are removed/returned at the end of round. But even tokens and markers remain if the hero is defeated. So does Pyro.