What is adhesive tape (or "Scotch" tape) that is used for packaging things is usually called in the United States? What verb do you usually use for "package something with packaging tape"? Maybe "to scotch something"?
This is called packaging tape, or packing tape. The verb used would usually be simply to tape. If the speaker wanted to be more specific, he could say "I taped the box shut with packaging tape".
"Scotch tape" is generally used in the United States for a much lighter variety of tape that is mostly suitable only for taping papers, and wouldn't be used for something like boxes. This is despite the fact that Scotch is a brand name of tape, and could include numerous tape products.
From Wikipedia, this is what Americans generally call "Scotch tape" (and what they would think of if they heard "Scotch tape"):
I should note that in American use, it might not always be capitalized and could very well appear as "scotch tape". I've even see "scotch-tape".
In the U.S. "Scotch" is a brand name of tape, manufactured by the company 3M. The Scotch brand is so prevalent in the U.S. that Scotch tape has become a genericized trademark.
As Jim said, when hearing the term "Scotch tape," most Americans will conjure an image like this in their minds:
However, 3M makes other kinds of packing tapes (sometimes called strapping tape as well, particularly when it's reinforced), also under the Scotch brand name:
Here's a screen shot from the 3M website (note the consistent use of the word Scotch for 3M tapes, and the several variants of packaging tapes available):
That's all I've got – I think that's a wrap.
I know OP asked about the US, but in the UK we call this stuff parcel tape...
...and this is how our eBay "winnings" arrive wrapped up in it...
Anyway, that's enough pictures. The more lightweight stuff OP calls Scotch tape is usually called sellotape in the UK. That's just another brand name - the actual 3M Scotch brand is probably more common, but we stick to sellotape for the generic name (as with hoover and biro, no-one cares that your particular hoover is made by Electrolux, or your biro by Bic).