There's a number of answers given here some of which are fine, others of which are not so fine for various reasons (even if they are correct), so I'll just answer each of the questions in order. Also, for clarity (and because it's easier to figure out the opposite from this), I'm going to assume in each situation it's 2 different players playing the original spell and the radiate.
1. Do I own/control the copies? If I Radiate a Lightning Helix, will I gain life from each copy?
You control each of the copies, since you created them. You'll gain life from each copy.
2. What happens when I Radiate a Bribery in a multiplayer game? Does "opponents" refer to MY opponents, or the opponents of whoever cast the original Bribery?
This one is a bit tricky. Radiate looks at what the original spell could target, not what the copies could target. So, if the original is not targeting you, you'll get copies targeting each player not targeted by the original (except the player who played it), including a copy targeting yourself. The copy targeting yourself will be countered on resolution because you are not a legal target for a bribery you control. In a single player game, there's only one target the original can target, so no copies are created at all.
3.a. If someone plays Agony Warp and has both effects target a single creature, does that count as a legal target for Radiate? If so, can I choose to split the targets of each copy?
It is a spell that targets only a single permanent. (It is not, however, a spell with a single target, should some other effect care about that.) See below for more on choosing targets for the copies.
3.b. If I Radiate a modal spell that has had its modes chosen such that it has only one target, can I re-choose the modes for each copy so that they have multiple targets? Specifically I'm thinking of Cryptic Command, here. If the original Command is bounce+draw, can any of my copies be re-chosen to be bounce+tap? Could I choose tap+draw, which still has one target, but not of the same type as the original spell?
3.c. If not, can I re-choose the modes as long as each copy still targets the thing it was originally targeting, but does something different to that target? Specifically I'm thinking of: there is a Grizzly Bears and a Brass Man in play. Someone casts Bant Charm, choosing to put the Bear on the bottom of library. If I Radiate targeting Charm, can I choose the copy targeting Brass Man to destroy the Brass Man?
3.d. Same situation, but the player originally casting Charm targets the Brass Man and chooses the "destroy artifact" mode. Will I get a Radiated copy targeting Bears, in the mode that can apply (bottom of library)?
Copying a spell copies choices made when playing the spell, including targets and modes. Most spells which copy spells specify that you may choose new targets for the copy (see fork, twincast, etc.). None say you may choose new modes for the copy. Radiate doesn't say anything about any choices concerning modes or targets, so the player doesn't have any choices. The mode(s) chosen is copied, and each copy created has it's own target which the player doesn't get to change. Since there are multiple copies being put on the stack at once, the player can choose what order they go on the stack however (and by extension what order they resolve).
