Stage 1F reads:

Forced: At the beginning of the round (before the resource phase), place 1 resource here. Then, if there are 4 resources here, advance to stage 2.

Stage 2F reads:

Forced: At the beginning of the round, add 1 set aside Host of Angmar (from any stage's set aside area) to this staging area.

Am I correct in assuming that you do not put a set aside Host of Angmar into play on the round you move from stage 1 to stage 2, since you would trigger all "beginning of round" effects once only, and Stage 2F was not in play at that moment?

Stage 1F reads:

Forced: At the beginning of the round (before the resource phase), place 1 resource here. Then, if there are 4 resources here, advance to stage 2.

Stage 2F reads:

Forced: At the beginning of the round, add 1 set aside Host of Angmar (from any stage's set aside area) to this staging area.

Am I correct in assuming that you do not put a set aside Host of Angmar into play on the round you move from stage 1 to stage 2, since you would trigger all "beginning of round" effects once only, and Stage 2F was not in play at that moment?

I asked Caleb that question at Gen Con and he confirmed this assumption.

That's weird... when checking for forced effect at "end of the round" we can have one that add another forced effect at the end of the round (Saruman ally) and we must resolve it because it is "still" the end of the round... Here, we are still at the begining of the round and I don't see why it should be otherwise. One possible answer would be that there is something between 1F and 2F I don't know of because I don't have the cards.

That's weird... when checking for forced effect at "end of the round" we can have one that add another forced effect at the end of the round (Saruman ally) and we must resolve it because it is "still" the end of the round... Here, we are still at the begining of the round and I don't see why it should be otherwise. One possible answer would be that there is something between 1F and 2F I don't know of because I don't have the cards.

Can you paste that Saruman ruling here? The difference might be that Saruman's passive effect triggers before Forced effects?

And all the cards for Annuminas are on Hall of Beorn.

Q: Here, we have a Cave Entrance in the staging area, and we have played Silvan Refugee and Saruman, targeting the Cave Entrance. The end of the turn arrives, and Saruman leaves play.
We are now dealing with a Forced effect from Silvan Refugee that trigger on Saruman leaving play, and a Forced effect from Cave Entrance that trigger on end of turn but could not have been resolved before Saruman left play. What is the right order of resolving this?
A: At the end of the round, you discard Saruman. That immediately triggers the Forced effect on Silvan Refugee, discarding it from play. Now it is still the end of the round when Cave Entrance is considered to be in play again, so you would need to trigger its ability before beginning the next round.
http://sdajce.forumactif.org/t2254p15-question-effets-en-fin-de-tour-saroumane-entree-de-la-caverne-refugie-sylvain-reponse-officielle (French)

I just emailed Caleb and he said I was correct.

Beginning of the round is a much neater concept than End of the round for these sorts of effects, because it's clear when things happen. End of the round is unpleasant because the "end of the round" effects trigger before the round has properly ended.