Move file_handlers API from //chrome to //extensions

Largely mechanical CL via move_source_file.py, subbing some Profile usage
for BrowserContext.

Results in removing some DEPS exceptions and #includes from //apps.

BUG=679870

Review-Url: https://codereview.chromium.org/2685453002
Cr-Original-Commit-Position: refs/heads/master@{#449460}
Committed: https://chromium.googlesource.com/chromium/src/+/10d5e4e8430e41ca148452f2f8582db517846799
Review-Url: https://codereview.chromium.org/2685453002
Cr-Commit-Position: refs/heads/master@{#450144}
29 files changed