old (collapse) the structure while I’m at its closing element. How can I quickly do this?

View > Fold Current Level (Ctrl+Alt+F) (use the shifted version to unfold, or checkmark Settings > Preferences > Editing 1 > ☑ Make current level folding/unfolding commands toggleable to make it a toggle, so the same keystroke folds and unfolds the current level). This will work whether it’s XML or HTML or JSON or anything else with folding.

Even though it has “level” in the name, it only does it for the current group, so in

<one>
  <two>
    <three/>
  </two>
  <anotherTwo>
    <anotherThree/>
  </a|notherTwo>
</one>

if the caret is at the | in </a|notherTwo>, it will only collapse <anotherTwo>...</anotherTwo>, not anything at level 2.

To answer your subquestion, though it’s not as important, there is also a way to:

quickly navigate to the matching start element and fold from there.

@rdipardo’s “HTML Tag” plugin has the Plugins > HTML Tag > Find Matching Tag feature (defaults to Alt+T), which does that aspect (even in an XML file, even though the plugin has “HTML” in the name). (You can install the plugin through Plugins > Plugins Admin)