In tonight's Ranger game, a Ranger was checked into the Flyers net. No penalty was called and the net was NOT dislodged. If a Ranger shot crossed the goal line with the player still in the net, assuming he did not interfere with the goaltender, would a goal have been awarded?
Basically, the answer is "maybe", but by-the-books and if you assume the referee agrees with you exactly, then the goal is allowed.
Strictly speaking, the presence of an attacking player in the crease (whether across the line or not) does not directly determine if a goal is allowed. See Rule 69.1:
69.1 Interference on the Goalkeeper - This rule is based on the premise that an attacking player’s position, whether inside or outside the crease, should not, by itself, determine whether a goal should be allowed or disallowed. In other words, goals scored while attacking players are standing in the crease may, in appropriate circumstances be allowed. Goals should be disallowed only if: (1) an attacking player, either by his positioning or by contact, impairs the goalkeeper’s ability to move freely within his crease or defend his goal; or (2) an attacking player initiates intentional or deliberate contact with a goalkeeper, inside or outside of his goal crease. Incidental contact with a goalkeeper will be permitted, and resulting goals allowed, when such contact is initiated outside of the goal crease, provided the attacking player has made a reasonable effort to avoid such contact. The rule will be enforced exclusively in accordance with the on-ice judgement of the Referee(s), but may be subject to a Coach’s Challenge (see Rule 38).
So - if, in the judgement of the referee, the presence of the attacking player did not have any effect on the goalkeeper's ability to defend the goal, the goal would be allowed. The fact that he was pushed can be relevant if there was contact with the goalkeeper, but wouldn't really matter in the case you explained where there was no contact. If he had been contacted, though, it might be allowed or disallowed depending on the situation.
I would underline, though, that this is in the judgement of the referee, and they tend to disallow goals fairly aggressively. In order for the attacking player to have no impact on the goalkeeper's ability to defend, it would require the goalkeeper to be fairly far out of the crease most likely, for example - and the puck would have to come from a direction that the player wasn't possibly blocking the goalkeeper's vision. If any of those are not true in the view of the referee, then they will determine whether the attacker made a reasonable effort to get out of the situation - again, if they did then the goal is allowed (as they were pushed in), but if the referee doesn't believe they attempted to immediately leave the area, they may disallow the goal.
There is a table in the rulebook, on page 154 (note: this is by the page numbering on the pages, not the PDF's page numbering, which is 165 in my browser), that explains the different situations. The first example matches your example, and says the goal counts.