Issue 1. I do both. Puppy has her own training sessions, separate from everyone else. But one of my favorite things to do is to also do some training while the others are loose around the house or yard. They show up pretty quickly and work for itty bitty pieces of food too. (Dh says it looks like I"m "holding court" because I'm surrounded by three very attentive dogs!)
When I was first working on Down with my pup, she was backing into the Down (I was putting the food between her paws to lure it) then sort of rolling to her side. Then the Big Kids showed up. After watching them drop into an immediate Platz a couple times, a lightbulb went off, and sure enough, she pops into a Down like the others. Pups just learn fast when there's someone that they can watch and learn from.
Also my kids are competitive. Whoever gets into the Sit, Down or whatever first gets their treat first. The pup actually has the advantage, since she's so low to the ground. So it's good for her confidence. But it sharpens the other two as well. They respond faster. On Wait commands (when I return to them), I reward based on seniority within the pack. But on Come, it's whoever gets there first. Everyone races to get to me!
Like Tracy, I use names a lot. When I call them, I use each of their names individually. Other times, I tell the Camper to "Find Meri!" and he herds her back to me. I tell the puppy to "Play with Camper!" and she bounds off in search of her GSD brother. And when Zamboni is off in the back corner of the yard, I ask "Where's Zamboni?" (A common command at my house since Boni's hearing isn't great), which means "go find her and bring her back." It's the job of all the younger dogs to get Zamboni and bring her back in when she's diddling around a bit, but can't hear me calling.) Learning each other's names (and ours) is really useful in a 5-member family. Training them together helps solidify that they're a team as well individuals.
Issue 2. It depends on the treat. If I'm giving a chew, the puppy eats hers in her crate. Otherwise, big brother finishes his in no time and starts stalking hers. If they're the same small treat, then they line up and give me a nice Sit or Down to get their treats. No one gets a treat in my house until they earn it. Everything is part of training.
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Issue 3. Because I do train together daily, my kids don't seem to get too jealous of each other. The Big Kids seem to see the new kid a way to get even more treats. I make sure that I spend plenty of time with the Big Kids. When I enroll the puppy in puppy classes, each of the Big Kids gets his or her own night as well. Either they're signed up for their own class or they get a committed night where I take that dog (and only that dog) to the park -- and I mix up which park as much as possible -- or other location every week. One of my trainers calls it "Mommy and Me" time. When I'm playing with the puppy with toys in the living room, my senior is sitting next to me, getting petted; and my GSD is chasing a Cuz that I'm tossing down the hallway. I stop petting long enough to toss the Cuz, and go back to petting. Meanwhile, I'm still flipping around a stuffed toy for the puppy to attack and play with with my other hand (the problem with three dogs -- not enough hands!).
So everyone feels like they're getting the kind of attention they want. After we do this for a while, we go outside, the GSD goes outside and we play soccer while the puppy is in the outside x-pen, and my senior sniffs around and jogs back and forth with him. Once the game slows down quite a bit, the puppy comes out and chases him and the senior is content to lie on the cool grass and enjoy a nice summer evening. It's multi-tasking, plain and simple (and exhausting!)
My first thought when you say your other dog is feeling jealous is that in actuality, he's not getting as much exercise as he used to be. I'd start there. When we're juggling puppies and adults, sometimes, we slip on the stuff that was second-nature like making sure our dogs got plenty of physical exercise (and perhaps mental stimulation). I try really hard to keep the Big Kids on their usual exercise/walk routines. It can be hard (esp when I've been up with the pup a lot the night before). But if you can, that's where I'd start.
