Kinda. Have a look at the
auto_complete setting in Preferences > Settings - Default. Copy what you want to change to your User settings and edit it there.
The basic problem you're having is that they syntax for HTML files is "text" rather than "source", where autocomplete is constantly triggered. You can change this behavior by adding something like:
"auto_complete_selector": "text.html - text.html.markdown, source - comment, meta.tag - punctuation.definition.tag.begin",
This adds the HTML scope to trigger auto complete (but not for Markdown).
To auto close tags you want to set:
To detect open tags I use the Tag plugin's Lint feature. (There are probably other alternatives.)
Please note that: autocompletion behavior may be different depending on the plugins you've installed and also that I did not test the above -- but it should work
Hope some of this helps,