The last time Britain won five medals at the event was in 1961, when the team returned from Belgrade with two gold and three bronze.
The British trio will be joined in the finals by Irish pair Paddy Barnes, a bronze medallist at the Beijing Olympics, and Darren O'Neill.
Khalid Yafai, who competed in Beijing, gave a brilliant display to shut out his rival, who beat the Birmingham fighter in controversial fashion at least year's World Championships.
Behind the scenes with Britain's top amateur boxers
Yafai said: "I wanted that revenge so badly and there was no way I was going to get beaten on my birthday. I just wanted to prove everyone wrong and make sure I got into the final."
Yafai's final opponent, Russian Misha Aloyan, beat him in the European junior final three years ago.
"We're the best two at the weight and it will be a good fight," added Yafai. "But I've matured and I'm very confident of beating him."
Southpaw Weaver, who is in his first year with the elite GB 'podium squad', started the last round level but pulled away to score a 10-3 win and set up a final against Denis Makarov of Germany.
Weaver has excelled after being fast-tracked into head coach Rob McCracken's podium squad and he said: "Nobody really expected a lot of me out here but I always really believed I was one of the best.
"I don't really notice the pressure, I just get on with the job. There's a great team spirit and it says a lot about how far we are all progressing that we get three in the finals of the toughest tournament of all of them.
"If you'd told me two years ago that I'd be boxing in a European final this soon maybe I wouldn't have believed you. But now I've got here I feel like I belong and I'm confident of bringing home the gold."
Stalker, who beat world champion Domenico Valentino of Italy in his first fight in the tournament, recovered from a final-round deficit to win 5-2 and secure a final against Albert Selimov of Russia.
Eighteen-year-old Gamal Yafai, in his maiden major championship as a senior, put up a superb display against Russia's world bantamweight silver medallist Eduard Abzalimov but dropped a 2-1 lead in the final round to lose 3-2.
Barry bantamweight Selby was unlucky to suffer a 5-0 defeat to Ukraine's Georgi Chigaev and had to settle for his second consecutive bronze in the event.
Belfast's Barnes secured Ireland's first final place with a comfortable 6-3 win over Spain's Jose de la Nieve at light-flyweight, while middleweight O'Neill beat Bulgarian Mladen Manev 6-0.
But three Ireland team-mates failed to join them. As well as McCullagh, Eric Donovan was beaten 8-2 by Russian Albert Selimov, while Beijing silver medallist Ken Egan blew a 7-2 lead to lose 11-9 to Abdelkader Bouhenia of France.
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