Whew! It's done!
As I understand it, Plucky was trying to do his homework Sunday night, but fell asleep and woke up Monday morning when it was too late to finish it. He then skipped a week of school to build the time machine, (becoming 'future Plucky') and traveled back to the beginning of the week to Monday morning to pick up 'present Plucky'. They then both went back to Friday night so 'present Plucky' could do the homework.
This is a bit of a paradox, because they should have met a 3rd Plucky, (past Plucky) on Friday night, but he wasn't there. Maybe he wasn't home? Or maybe they just didn't want to complicate the story too much.
It's probably a bit much to believe that 'past Plucky' would have actually been home to do his homework on Friday night anyway, but if he would have come home then he would have found 'future Plucky' there.
So back to the chronology. The two Pluckys are at Friday, and 'present Plucky' steals the time machine and goes back to Monday morning to show Buster and Babs. They all then go back in time, Plucky gets lost, and Buster and Babs end up in medieval times. Babs rescues herself and Buster from the dragon, and then they go in search of Plucky. How they manage to find Plucky is a mystery, but it's a cartoon, so I just ignore it.
'Present Plucky' steals the time machine again, stranding Buster and Babs in prehistoric times, and goes back to the present bringing the prehistoric Dizzy with him.
I'm not certain when 'present Plucky' arrives this time. I think he ends up in his pond, or just a pond, which waterlogs the time machine, and as far as I can tell he doesn't leave the area until the machine is working again, so he doesn't interact with anyone that we know of. Dizzy interacts with Elmyra, but there's nothing to indicate a time for when this happens, so there's no paradox created in this instance.
'Present Plucky' and prehistoric Dizzy travel back to prehistoric times where Buster and Babs have saved Acme Acres. Then Buster, Babs and Plucky all go back home to sometime Sunday, not sure exactly what time Sunday, but it doesn't matter. They find 'future Plucky' there, but not 'past Plucky'.
'Future Plucky' then takes the time machine back and disappears, presumably back to his present, which is later in the week, but we don't know that for sure. Knowing Plucky, with access to a time machine he may have gone sometime else to try and become famous and wealthy, the same way 'present Plucky' tried to do.
So, at the end of the episode, 'future Plucky' is gone, 'past Plucky' is nowhere to be seen but must still be around somewhere, and 'present Plucky' is left to do his homework again. This means there are still two Pluckys, and there should also be two sets of Buster and Babs until Monday morning.
Come Monday morning, 'past Plucky' would become 'present Plucky', and 'present Plucky' would become 'future Plucky' who would then spend the next week building the time machine so he could go back to Monday morning to start the process over again. Amazingly, this whole thing works out fairly well as long as certain things happen.
Here's how I see it working out. We'll follow Plucky from his perspective. First, starting Friday after school, Plucky goes to Hampton's house for a weekend-long slumber party with Buster, Babs, and other students from Acme Loo. The party doesn't end until late Sunday afternoon. Plucky then goes home and remembers that he hasn't finished his homework. He starts on it, but falls asleep until Monday morning. His future self shows up, takes him back to Friday, and he steals the time machine. He goes through the whole storyline of the episode, finally ending up back at Sunday sometime before he got home from Hampton's party. His future self takes the time machine back leaving Plucky to do his homework.
Plucky realizes that he has an extra day to live over, though, and doesn't want to waste it doing homework, so he goes to an all night video arcade where he plays video games until late Monday morning. After all, he figures he's learned his lesson and that his other self will do the homework when he comes home from Hampton's party. He comes home from the arcade Monday morning, finds no one there and his homework still not done, and realizes he's now late for school as well, so he figures if he can do it once, he can do it again and spends the rest of the week building his time machine which he finishes the following Saturday.
He goes back in time to Monday, takes his past self back to the previous Friday thinking he'll trick him into doing the homework for him. He figures it'll be easy to trick him, since he knows that his past self will try to steal the time machine and thinks he can stop him, but he forgets how quick he can be and looses the machine again. He now has nothing to do but sit and wait for himself to get back on Sunday, so he plays his Nintendo until then, only looking like he'd been doing the homework just before his past self was scheduled to arrive. Once his past self is back, he grabs the time machine and heads back to his present, which is next Saturday.
Here's a graphical representation of the timeline. The blue lines indicate normal time, red are periods of time travel. All the episode stuff in pre-historic times has been pushed out of the left side to simplify things. Purple dots are nodes when time travel occurs.
So, as you can see, from Friday afternoon until Monday morning, two Plucky's exist, and for a brief time both on Friday afternoon and then again on Sunday, three Plucky's exist. It's a wonder the universe didn't explode.
As for Buster and Babs, they're much simpler. Two sets do exist from Sunday until Monday morning, but they were probably smart enough not to interact with their past selves. They probably showed up at school just after their past selves left with Plucky and continued their school day as normal so no one would know what had happened. It probably went down something like this.
"Hey Babs? It's Sunday, right?" Buster asked.
"Looks like it." Babs replied. "So?"
"So right now we're still at Hampton's party," Buster explained. "We won't get back home for another couple hours."
"Hey!" shouted Babs. "That's right! There's two of each of us in Acme Acres right now! We should go say hi!"
"No!" Buster exclaimed. "That could disrupt the timeline! Who knows what could happen? The whole universe might short out, or something!"
"Hmm..." Babs thought. "So we can't let anyone see us, and no one would come looking for us if we disappeared for a while..." A devilish look spread accross Babs face, and she put on her sultry voice. "Are you thinking what I'm thinking, Buster?"
Buster was taken aback by Babs' sudden change. "Err... Umm..."
Babs continued, using her finger to caress Buster's chin. "Maybe we should go somewhere quiet, you know? So we can be... alone..."
Buster, now dripping sweat, thought fast. "...And play video games until our thumbs hurt and our brains go numb?"
"Actually," Babs replied, now returning to her normal voice. "I was thinking Doogie Howser marathon."
"Well, I guess that's mind numbing enough." Buster groaned.
"Oh come on!" Babs retorted. "He's cute!"