Small yellow plastic part, shaped like a tootsie pop sucker

The little plastic part from inside of the pan gets thrown away. Ford puts them an most of their vehicle's tranmission pans, when tranny is serviced for first time it should be removed.

Great picture, nice question. I asked the same thing the first time I saw one of them. It is a plug that they install from the transmission manufacturing plant to seal the trans. It goes where the dipstick tube seats in. They install the plug and when it rolls down the assembly line, whoever puts in the dipstick tube just pushes it in, sending the plug into the trans pan. It will never hurt anything, but it makes you waist a lot of time looking for someplace to put it. We use to call 'em butt plugs, cause that's all they were good for when ya found em. Hehehe!
As for your trans not working it has nothing to do with the plug. You may want to inspect the filter and make sure it is seated correctly and the fluid level topped off. If you have any other questions feel free to give a shout anytime.
Dave

Is this a C4 tranny? I forgot the name for that part

That one goes to the rounded corner of the tranny housing the round part when install will be facing downwards to oil pan.

That is nothing to worry about. When the transmission was built, that was placed where the dipstick tube presently is. It was there to keep dirt out during production and when the tube is installed, it is pushed into the pan.