In Vim, is there a way to paste text in the search line?

Copy:
1) v (or highlight with mouse, in visual mode)
2) y (yank)

Paste:
1) / (search mode)
2) Ctrl + R + 0 (paste from yanked register)

Copy it as normal, then do CtrlR" to paste. There are lots of other CtrlR shortcuts (e.g, a calculator, current filename, clipboard contents). Type :help c_<C-R> to see the full list.

You can insert the contents of a numbered or named register by typing CTRLR {0-9a-z"%#:-=.}. By typing CTRL-R CTRL-W you can paste the current word under the cursor. See:

:he cmdline-editing

for more information.