How to cut a hole in a globe without breaking it?
I think you might be close to a solution by using heat, but consider to combine a source of heat with a cutter blade. This will allow you to use the tip of the blade to have a minimal impact on the surface, while providing the means to puncture the surface. You may find that you can re-heat the blade for each cut and use this method for the entire opening.

Your idea of repeat cuts is also good if you combine it with heat. This will minimize the damage to the surrounding area, while "gently" imposing the blade into the plastic surface.

The size you reference should be small enough to allow the rest of the sphere to maintain its shape. If an edge of the cut is snagged by an envelope, it may buckle inside, but the overall small size should allow it to pop back out.

With Henry Taylor's implied permission and recommendation, I am incorporating some of his support material below:


  Manually heating the blade is absolutely an option, but there is also
  a specialty tool called a "hot knife" which is just a soldering-iron
  with an xacto-blade as the tip. I find that using a hot knife to
  "scratch" at sheet plastic slowly with very low pressure and going
  over the line a dozen or more times usually provides the best results.
  Practice on less precious objects of similar material before hand.
  
  As for cutting the straight lines of a rectangular opening, set up a
  guide for your blade using layers of blue painters tape carefully
  applied around the outer edge of your cutting lines. Slide the blade
  along the tape wall with the tip resting against plastic at the cut
  point. Then once your hole is done, wet the tape to loosen its glue
  and peal off VERY CAREFULLY starting at the edge of the tape furthest
  from the hole with force perpendicular to the cut. Remember that you
  have seriously compromised the plastic's integrity by cutting it, so
  treat it like an empty egg shell... delicately.