New cat is unsettling the household

My family has had 3 cats for the past 12 years. (Archie, 15, male, friendly and very much the king; Emme, 13, female, skittish but affectionate; Maple, 12, male, independent and easy-going) My mother was always the primary care-giver for the cats and was always home with them, but she passed away in August. I left for 6 months of school soon afterwards and Emme died of multiple health problems in January. When I returned, Maple's personality had changed so that we was much more affectionate and needy and he slept with me every night. (He had never experienced any environment changes such as moves, adoptions and the like as Archie and Emme both had, so I understand that these changes greatly upset him) I have been home for 3 and a half months now, and Maple was mostly back to normal, although he was more open to displays of affection from me than he used to be. My dad and I decided that the cats were settled enough that we could adopt another cat. 2 weeks ago, we brought home Sashi, a 2 year old lynx-point siamese (although she's not a pure bred because she is missing some of the necessary markings) who is incredibly cuddly and loving (she's quickly a favourite with anyone who meets her). We understood that we'd need to take the introductions slowly, so she spent 4 days in my room with her own litterbox and food/water dishes, but she quickly went stir crazy and so we decided she was comfortable enough to let her out and meet the other cats. However, it was not her we needed to worry about. Archie was not fond of Sashi, but he put up with her, although avoided her if possible. Maple, on the other hand, hissed and growled whenever he saw her and would run away every time he lost one of their staring contests (which he most often did). Since Archie and Maple are both indoor/outdoor cats, we would let them retreat outside to territory that was completely their own whenever she became too much for them. However, Maple began to spend more time outside than inside, and 3 days ago, he disappeared. It is not uncommon for him to spend 1 full day away from home in the summer, but 2 days was already a cause for worry, and now we are VERY concerned that he does not feel wanted or accepted in his own home! Another problem is that now that Maple isn't around, Sashi has begun to pick on Archie and he's begun to retreat outside as well and I don't want to scare him away from his own home either!
What have I done to my household!? I already love Sashi, but I dearly love my two boys as well and I can tell how upset they are now that they should have had enough time to adjust (according to the animal shelter). We will begin looking for Maple, but what can I do to make Archie know that he's still loved and wanted at home? And how can I help my cats to get along? Any help and suggestions are greatly appreciated. Thank you.

We definitely know the benefits of having indoor only cats. The problem was that Archie was an outdoor cat before we got him, so he is not content to live indoor only and can become unbearable (peeing on everything, climbing on tables and counters, clawing furniture, general bad attitude) if he isn't allowed outside. Maple, unfortunately, picked this up from him and we found it best to let them go out. (Sashi, too, is now very interested in the outdoors... but we don't really want to go that path, at least for a while yet with her) Of course, we do make sure that they are always up to date on their shots etc. and we don't deny that we like the way they keep the rodent population down in our neighbourhood.
No sign of Maple yet. We haven't been able to get a hold of the humane society either, so we're really hoping that someone has brought him in. I think we'll be making posters soon and putting them up around the neighbourhood, too, and hopefully someone has seen him.
I have also moved Sashi back into my room for the nights (with litterbox, food and water dishes) so that Archie isn't getting bullied quite so much and will hopefully feel more in control again. All in all, I really don't think we went through the whole introduction process properly and now we have to figure out some way to remedy the situation before it escalates.

Just as an update, Maple returned this morning as though nothing had happened. The hissing and growling has resumed, but he seems to have a better attitude despite everything. Fingers crossed that he's decided to stick around now.

Thanks. I'll keep that in mind. We're going to give it another week and see how things go now that Maple seems to have decided to stick around (thank goodness!). Things are settling slowly as far as we can tell, so fingers crossed the next week goes well.