As far as I know there's not a definitive answer on the Lucky Egg, in large part because it's not considered to be likely to have an impact; apart from the coincidence in naming, anyway, it has no reason to be related. So I don't believe there is a study on the matter.
The Silph Road Reddit has seen quite a lot of research on the matter of Lucky Friends, of course; see for example this thread which used just over 1000 interactions to test various possibilities.
- They found that all four methods were roughly equal in probability; that is, Trades, Gifting, PVP battles, and Raiding, though they didn't necessarily do this scientifically.
- They found about a 2% rate overall (1/52), with fairly wide variance (the three of them had respectively 1/83, 1/39, and 1/46 rates).
This thread has somewhat larger numbers, and found a lower rate (around 1.1%). It's possible (likely?) that the rate decreased over time, as the first thread was much older, and Niantic regularly updates rates for things like this.
One person did post some stats suggesting Lucky Eggs boost Lucky Friends, and if those stats were reliable might be reason to believe it's true - but it's hard to say, both as it's unclear how reliable this person is and how carefully done the study was on their end.
As far as what does contribute, the only known things are the four interaction types - PVP, Raiding, Trading, and Gift opening - and only the first time per day (the first of any of those four, not separate for each); see the announcement article:
Grow even closer to your Best Friends in Pokémon GO with the new Lucky Friends feature. Once a day, when you interact with a Best Friend by battling, trading, opening a Gift, or teaming up in a Gym or Raid Battle, you have the chance to become Lucky Friends. Then, the next time you trade with each other, both Pokémon will definitely be Lucky Pokémon—coveted for their great stats and lower Stardust requirements for leveling up. Trade a Pokémon you want to make battle-ready so you and your friend can both be better prepared for big battles to come!
Once you've completed the trade, you'll return to being Best Friends again. Try again the next day for another chance to become Lucky Friends.