How does one escape backslashes and forward slashes in VIM find/search?

Same way you escape characters most anywhere else in linuxy programs, with a backslash:

:%s/<dog\/>/<cat\\>

But note that you can select a different delimiter instead:

:%s@<doc/>@<cat\\>@

This saves you all typing all those time-consuming, confusing backslashes in patterns with a ton of slashes.

From the documentation:

Instead of the / which surrounds the pattern and replacement string, you
can use any other single-byte character, but not an alphanumeric character,
\, " or |. This is useful if you want to include a / in the search
pattern or replacement string.