Couple of comments if I may (coming late to this thread), background, single dad second AP just arrived, first was with us jsut for the summer.
When AP1 arrived DCs were all over her for about a week and then as others have said it the novelty started to wear off and a sort of equilibrium was reached after two to three weeks.
As a single dad I initially felt very awkward having this extra person in the house, and really barely new what to say beyond talking about some very basic facts of what had happened that day (no doubt some MNers will say they this a common problem with men and not restricted to when they speak to APs) but again things improved as time got on.
With regards to discipline I have said that if in doubt they are to side with the kids (e.g. but dad always lets me ...) BUT to report any doubts to me and I will deal with them. They are very much to try and be the dcs friend - I see the relationship very much as big sister. AP1 came to me about four weeks in and apolgised that she had to raise her voice to ds1 that morning, she then described what had happened, given that I would have probably have been throttling him at that point I congratiulated her on dealing with it correctly and then went in search of ds1 who was already in shock and chastened at raised voice.
With regards to the weekend I have been very clear with both from the start that this is there time and that they should sleep in and use this time to get out and explore. That they are welcome to eat with us when we are eating or they can just help themselves from the kitchen. If we are going out somewhere that interest them they are welcome to join us but I don't expect it of them.
I think others have covered off timetable, but both have had a timetable as part of their welcome to the house routine. It is quite prescriptive which has lead to its own comedy moments when ds1 saw it and then a few days later refused to move towards the bathroom five minutes before the prescribed time in the evening.
The one thing I haven't seen on here (and I confess to having speed read it) is how have dcs reacted to her and what would they feel if she suddenly disappeared?