Illustrator only just recently (CS3, I think) gained a limited ability to split multiple paths at selected anchorPoints. It's as awkwardly-interfaced as most aspects of Illustrator. It's a button in the Control Panel concisely named "Cut Path At Selected Anchor Points" (deep breath).

If you don't see the button, it's because you have to select the point(s) you want to split with the (deep breath) Direct Selection Tool (white pointer). Illustrator's Control Panel can't decide if it's tool-based or selection-based, so it often hides the features you need for what you are doing.

If you still don't see the button, it's because you logically have all the anchorPoints of the path selected when you want to split all the points of a given path. This illogical abberation is because, despite Illustrator's insistence upon tedious multiple selection tools, it still is not smart enough to understand the difference between a path being selected as an object, as opposed to merely having all its anchorPoints selected.

The button does appear, however, in the "opposite" situation: if you "select the segment" (not the points) of a simple 2-point path. In that situation, the button appears, but of course it does nothing. But it's there, in case you want to do nothing.

So, as with most things in AI, if you want to cut all the anchorPoints of a single selected path, you have to come up with workarounds:

Select the path, then DEselect one anchorPoint. Now the button will appear. Then cut the other anchorPoint.

Or, use the workaround that pre-dates the new feature by many years: Use the Select>Object>DirectionHandles command. Then Cut. Then Delete. Then Paste In Front. That completely screwey routine probably needs some explanation:

The command says you are selecting "Direction Handles". Elsewhere, Illustrator's interface and documentation pretend that you can select "segments" independently of their anchorPoints. Of course, what really gets copied to the clipboard are individual 2-point paths (each including 2 anchorPoints, 2 handles, and the segment they describe), and that's what you get when you Paste thereafter. But when you do the Cut, the anchorPoints are "left behind", because they ostensibly were "not selected." That's why the Delete command in the workaround, before Paste In Front--to get rid of the stray points that are left behind when you copy them to the clipboard by selecting Cut. (Clear?)

Note that when you ostensibly "select the handles", the display is exactly the same as if you "direct selected" one "segment" with the (deep breath) Direct Selection Tool. In such a case, you often can't tell by looking which "segment" is selected. You just have to remember what you clicked or selected and with what tool or command.

As for joining multiple paths at once, Illustrator has not got around to that yet. (It only took 20 years to gain the above-described ability to cut multiple selected points.) Workarounds:

Use a 3rd-party plug-in. This assumes, of course, you can stomach the principle of paying exta above-and-beyond the bloated price of Illustrator for such basic functionality.

Or, use a free script. You are welcome to one or both of mine if you wish. JET_JoinNearest respects pre-existing outboard handles (if any). JET_JoinNearestStraight joins with straight segments.

Or, assuming the series of paths together visually describe an enclosed area, you can use the flood-fill feature (LivePaint) to convert the paths to a "LivePaint Group", use the LivePaint Bucket Tool to create a new closed path, then Expand and Ungroup the LivePaint Group to get back to normal basic paths.

Welcome to Illustrator.

JET