My girlfriend and I have been playing regularly at our local store, and have been there for almost all of HotDQ Ep 1. However she had to miss the last session and, that being the one where we finished the episode, she missed out on receiving the gold, downtime, renown, and XP.
Now some of those rewards (gold, downtime) aren't crucial, but is she gonna be constantly one level behind going forward?
This doesn't sound right to me. If the player was present for any of the sessions they should earn appropriate XP and treasure for those sessions. There is a max of 300 XP for episode 1. So if you hit that (which many of our players did by the end of the second session) then missing a week shouldn't be a big deal. If they're not present for the session when the episode ended I'd still award them downtime days, although I might not award full days. It depends. For example, if episode 1 took four sessions and you were there for sessions 1-3 then I'd give you the full reward. If someone was only there for session 4 but missed the first three I wouldn’t necessarily give them the full downtime day reward. DMs call, I suppose.
Regarding treasure, PCs kind of get screwed for episode 1 since any treasure they find is technically loot the Kobolds and cultists plundered from the villagers. So any party of good PCs should feel strongly motivated to return it to the rightful owners, or at the very least give it to the Governor so he can help get it back to the owner. PCs who keep the loot may find it difficult to sell and may make enemies in Greenest if they try to claim the loot for themselves.
Yeah, talk to your DM. It seems the AL rules are pretty loosey-goosey regarding this, and it's being handled in different ways at different locations. (Which could be a plus from a certain perspective, or a negative, depending on how you look at it.)
As far as I personally can tell, there is no mechanism for officially awarding partial Downtime or Renown in HotDQ. The system currently doesn't seem to accomodate that--it doesn't really deal with "player churn" very well. You're best bet is to talk to your DM. Your location's Organizer may have feelings about it, too. At my store, the DMs defer to the Organizer's preference so it's the same for all the tables.
The consequence of not offering partial Renown and Downtime results in the implication that there are certain days when it's more important to show up for Encounter play than others so you can level your character. Which logistically is a really raunchy system, and is biased toward players who have committed to a long-term engagement. Encounters play is written for and encourages recurring weekly attendance, but in reality that is not the case for a lot of tables and locations, as you yourself have experienced.
Regarding leveling opportunities in Episode 2--it is not written and structured the same way as Episode 1. Again, the guidelines say characters can't level until the end of the Episode or a Long Rest. This implies that if the DM decides to allow a Long Rest at some point in Episode 2, it's possible that your girlfriend's character might have the chance to level up (depending on whether she's accrued enough XP along the way), but one is not explicity outlined in the Episode. If the DM hasn't contemplated allowing a Long Rest mid-Episode, it may be challenging for her to level up mid-Episode and she may have to wait until the end.
Which brings us back to my initial point--make sure you know when that date is if they're not awarding partial Downtime and Renown. It sucks, but it seems the knife cuts both ways--having that much DM/table flexibility can be good as well as bad. I hope your particular DM is accomodating, for your girlfriend's sake. 
XP and gold are rewarded per session. Downtime and Reknown are awarded at the end of the episode. This is in the HotDQ Encounters packet given to each Encounters DM.
Yes ,it would be possible for her to 'catch up' and hit level 3 with everyone else.
I was more concerned that this was yet another DM who was waiting till the end of the episode to do log sheets. I really cant understand that concept.
There is more than enough exp in the different episodes for characters to catch up.
Matthew
Just to clarify, character can level up at the end of each session, not just at the end of the episode or after a long rest. So if your character earned 250 XP in Episode 1 and then earned at least 50 XP in Episode 2, session 1, you can/should level up before Episode 2, session 2. I know it seems a bit silly from a story point-of-view, but that's what's written in the public-play PDF version of the adventure.
Just remember that there is a 600 XP max for Episode 2 so to continue with this example, the same PC who began Episode 2 with only 250 XP will hit level 2 after one session, but won't hit level 3 until after completing one or more sessions of Episode 3 since they'll fall 50 XP short of the required 900 XP for level 3. In this circumstance the player may want to play at least one of the D&D Expedition adventures to earn some additional XP.
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