"Can I come together with my friend?"
The characters we separated illogically into the blocks: 一起 is a word and 来 is a word, so why make blocks that say 一 and 起来? It's confusing.
It might be confusing but it also helps with word placement because it shows you that the 一起 (together) has to come immediately before the verb, so it stops you putting the 一起 in the wrong place.
Keyi可以 gives the idea of something being acceptable or allowed. Neng能 has the idea of possible, having the ability to do something.
No, they're interchangeable. I don't know why this exercise doesn't accept 能 as a correct answer.
The meanings of "能" and "可以" overlap; I'd expect "我能跟我的朋友一起來嗎？" (with "能") to also be accepted per the grammar at https://resources.allsetlearning.com/chinese/grammar/Comparing_%22hui%22_%22neng%22_%22keyi%22#Expressing_Permission.
That grammar indicates that "能" and "可以" are like "can" (expressing ability) and "may" (expressing permission) in English, respectively, and likewise interchangeable in circumstances such as this one.