I'm on my mobile but will try to cover your questions.
Futaba 2.4Ghz receivers come in two types, one with external antenna and one with internal, hence, no wire. It's common to see both types. You will see the RX without designated with an "E" in the part number.
Yes, you can cut your antenna tube down to size, just long enough to contain the wire. I do that all the time, looks nice .
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