Mapping Center Answer:

The whole topic of interoperability between ArcMap and Illustrator is, as you might imagine, an extensive one. I'll limit my answers to those you have posed here, but you should consider consulting other web-based cartographic resources for additional discussion. CartoTalk is one good source for finding practical information with regard to using Illustrator for cartography.

Off the top, there is an option in ArcMap from which you can export your map to four-color separates for printing. This requires that you have the ArcPress extension activated (this is included with all licence levels after version 9.0), and a connection (or the installed software driver) to a PostScript-enabled printer. Otherwise, changing your color space is not mandatory if your intent is to export to the AI format, as Illustrator manages color in RGB, CYMK, and other color models.

The most simple way to get your map out of ArcMap is to use "Export Map" from the File menu. ArcMap provides you with options to export to ten different formats, including Adobe Illustrator (*.ai).  ArcMap feature layers and Group Layers are respected on output, which includes separation of labels and annotation into their own Illustrator Layers.  As far as text management goes, pick one application or the other - I'd recommend that you stick with ArcMap, in that you have rule-based placement control over multiple labels, especially if you use Maplex.

The import of AI and PDF formats is not well supported with ArcMap, so a return trip is likely to have some issues.  A useful work-around is to export your data from Illustrator in a CAD format (DWG, DXF) and bring it back into ArcMap that way.  However, this is a poor, fractured, and very much NOT recommended practice.  Part of what we're trying to do with Mapping Center, and at ESRI in general, is remove the need for users such as yourself to have to leave the ArcGIS environment at any point in your map production workflow.