New location? If yes, what kind of setting would it have to be?
I'd approach this from the director's viewpoint rather than the fan's. “Special” locations, like the Kingdom of Zeal and the Dead Sea, which are “special” because they are tied to key plot events, cannot be called upon again. Their magic is closely tied to the revelation of these plot events, and, once those surprises are spent, the specialness passes into memory. I consider it a breach of artistic principle to try and live off the reputation of past story glory, so I would develop new special locations. However, many ordinary locations featured or mentioned in the first three games would definitely reappear in any game of mine, as would new locations. I'd expand upon the 1999 time period, and explore the possibility of new eras and dimensions.
From a thematic standpoint, I think I would conform more closely to Chrono Cross or Radical Dreamers than to Chrono Trigger. The latter depicted a very wide variety of settings—a true adventure across history—whereas the others focused more intently on a much narrower setting. More than anything else, I think this is what changed the tenor of the series, and it is a change I would probably uphold...although I might hybridize a wee bit. (See the question about time travel and dimension crossing.)
New cast of characters? If yes, what kind of characters would you like to see?
I firmly hold that each main sequence game in the series ought to have its own, new main character. Neither Crono nor Serge would be playable characters, let alone main characters, in my game. I would also be respectful of plot constraints from earlier stories, rather than trying to work around them. For instance, as much as I feel that her death was a waste, I would probably accept Lucca's implied death at face value and leave it unchanged.
I would simply ignore most of the Chrono Cross playable cast outright. Any returning characters from any of the three games would be significant, memorable characters. I would very likely not promote previously minor characters like Toma or Cyrus into playable roles.
Most of my playable characters would be new. Any returners would depend upon the requirements of the plotline. However, it would be nice to have at least one or two playable characters from both CT and CC. (Radical Dreamers has unfortunately been retconned out of the main sequence, so that's out, which is quite unfortunate indeed.)
I would depart from both the CT and CC cast styles in that I would have a larger playable cast, but with many of these roles being minor and/or event-specific. I'd also increase the maximum party size from three to four (or even five!), and feature more scenarios with parties smaller than the maximum.
New Lavos creature or a new version of Lavos? Or an entirely new enemy? Who should be the antagonist in this game?
No Lavos, in any form. Lavos was the ultimate Chrono series villain. Lavos cannot be topped with another even badder, meaner villain. I would take a page from RD and go in a different direction, by putting a human face on the antagonist. I'd like to have an RPG where the division between good and evil is not so clear cut. Taking a page from Hayao Miyazaki, I would render legitimatizing depictions of all “sides” of whatever conflict, and let the players decide for themselves who to root for.
Time travel and / or dimension crossing? Yes or no? Or something completely new? If yes, what?
I think the next main sequence Chrono game ought to make use of both time travel and dimension crossing. These are very different mechanisms; the existence of parallel dimensions in CC implied that more than one timeline can run simultaneously. That opens up the entire scope of travel into two dimensions: the T axis and the D axis. I see no reason not to incorporate both of these possibilities into the next game.
New gameplay or not? New battle system? New magic? Also let's hear any good gameplay ideas you have.
I'd de-emphasize the importance of combat, just as a matter of personal taste. I don't like it that game progress is designed to be achieved primarily and almost exclusively through the mechanic of killing. To me, this is a failure of the RPG genre in general. I'd undoubtedly keep a combat element to the game, because that would be in keeping with the Chrono series, but I'd also develop other mechanics.
I'm not gonna talk about battle and magic systems.
Who would have to draw the characters? Do you want Mr. Toriyama back, or Nobuteru? Or someone else?
This is a less interesting question to me; I really don't have a strong opinion.
Name the characters who would definitely, no matter what, have to make a cameo appearance. Can be anything, from playable characters to NPC's to Nu's.
Schala needs to be there. She has inadvertently became the main character of the series, so her presence is almost a necessity. I doubt I would make her playable, though. Gato, of course, goes without saying. So too do Serge, Kid, Crono, and Marle. I'd also revive plenty of minor characters for any time periods which get revisited. This would help to build continuity through the series.
Which time period has to be in it no matter what?
That's a deliciously open-ended question. It depends entirely on the plot. Based on the kind of story I'd want to tell, I think 1999 is the time period that has to be present. I'd also be open to 10XX.