When in XML mode, with the caret inside <a></a> like <a>|</a>, pressing ^< does not result in <a><p></p></a> as it does in TM1, but in <a<>></>></a>, which is not well-formed XML.
I had noticed this in HTML mode as well and I think I've found kind of a workaround. In the HTML bundle, select "Insert Open/Close Tag (With Current Word)" and change "Input" from "Selection" to "Nothing". This will prevent the behavior you are seeing but will also remove the ability to type an element name and invoking this function to make a tag out of it. TM1 seems to have more"Input" filter options.
Not sure what to do about your other issue with the ^<a completion. Maybe it is related to the input filters as well.
Hope this helps, Craig