Do adult games use delayed offside and allow icing when short-handed?

Yes to both. Adult classifications use the same rules as Youth 15-Only and older, girls' 16U and older, and high school.

Short-handed icing: a team that is short-handed (below the opponent's on-ice strength) when it shoots the puck may ice it without a whistle. This is judged when the penalty expires: if the puck was shot before the penalty time ran out, no icing is called.

Delayed offside: when an attacking player precedes the puck into the zone, the linesperson signals a delayed offside instead of stopping play. It's cleared if:

• the puck leaves the zone, or
• all attacking players are out of the zone at the same instant by touching the blue line (tag-up).

Play is stopped if an attacking player touches the puck, tries to gain possession or keeps pressuring the defending puck carrier. An intentional offside moves the face-off to the offending team's end zone.