Add them all to the library, then place them into the order you want them in.
Would I do File-->Import Movie?
After that, would I just drag them from the event library onto the timeline?
Also, is there any way to trim the videos?
Yes.
Yes.
Yes, in the timeline area.
This may help: http://etc.usf.edu/te_mac/movies/pdf/imovie09.pdf
What's happening is I import the video and I click on it in the event library, the Source Area shows "No Matching Video".
No idea ...
Maybe someone else can help.
I downloaded the clips from Youtube.
Should I try reencoding it in Handbrake and seeing if that helps?
Are they .flv or .mp4?
mp4 should work, but reencoding won't hurt.
If you re-encode them, also called "transcoding", be sure to use MPEG Streamclip, as it offers the Batch List function (Command + B) and is able to output .mov (Command + E, "Export to QuickTime") using the Apple Intermediate Codec (AIC) for video (leave it "unscaled" in the Frame Size section) and Uncompressed for audio.