Should Heimerdinger play burst or prolonged damage when he mainly wants to make the zone unplayable?

Most of the time he wins more through prolonged damage and zone control than through simple short burst. Heimerdinger mainly wants the enemy to pay for every second spent in his ground, not just the first well-landed projectile.

When is Crown actually better on Heimerdinger?

Crown becomes better as soon as dive or direct engage remove the very right to hold the zone. If Heimer can no longer survive long enough to let his turrets and slows work, adding only more damage stops making much sense.

Why is Rylai so often powerful on Heimerdinger?

Because it turns already annoying damage into a real pathing problem. With Rylai, Heimer does not merely hit: he slows the exit, dirties the entry, and makes the choice to stay one second too long inside his zone much more expensive.

Heimerdinger Wild Rift Best Build Guide (Patch 7.2a)

Heimerdinger does not seek direct one-shots; he seeks to stack unbearable pressure in an area he has already prepared. His default build must therefore reinforce damage that adds up over time, the ability to keep enemies within turret range and his survival margin once the fight finally reaches him.