I wrote a guide about everything that we know for farming- http://forums.spiralknights.com/en/node/108732

In this guide, I explain why you shouldn't be farming for eternal orbs of alchemy- because you find them statistically faster than radiant fire crystals which are needed to get the bulk of the power from leveling your gear to 5*, so you should just farm for those instead. As an example, I have over 150 eternal orbs of alchemy and a debt of around a couple thousand radiant fire crystals (i.e. items that I use and that require heating), all from farming for the fire crystals. The progress I make also provides me with considerably more radiant fire crystals than you would expect to get in the arcade or from Vanaduke, since all of the boxes that drop in Dreams and Nightmares are on depth 28, whereas the ones you'd expect to receive would range from depth 19 to 28.

And to my knowledge no one has really researched the issue of whether party size effects drop rate.. however our spidey senses tell us that it doesn't, as does Spiral-Spy's presumably data-mined (or invented?) blog on x-mas odds. Neither of these things are credible sources but I'd still say that they aren't unlikely. If you'd like a more concrete answer, do 100 runs on some level with people and without people, recording the number of crowns, orbs, fire crystals, and 5* materials you get and I'll run it through an online stats calculator and give you a hard answer for all of the above.