Did Kakashi ever love Rin or at least feel anything for her?
As you already mentioned in your question, 


  Mangekyo is, indeed, initially awakened by the trauma suffered from witnessing the death of someone close to the user. However, Uchiha have historically misunderstood this criteria as the user needing to be responsible for the person's death, and for that reason they developed the practice of killing their closest friends in order to gain the Mangekyō Sharingan.


But, it depends upon each individual and the amount of trauma felt by him/her whether the Mangekyo is "completely" activated immediately, as in the case with Obito, or after some time, as with Sasuke and Kakashi.

Obito loved Rin deeply, and thus the trauma suffered by him, on watching her die, was catastrophic. Thus, he "completely" activated the Mangekyo immediately. Rin was not just a team-mate of Kakashi but also a dear friend to him who remained with both Obito and Kakashi since their childhood.  So Kakashi too had a good bonding and a feeling of friendship with Rin, but he did not love her in the way as Obito did. That's why, Kakashi too activated the Mangekyo at the time Obito did, although "momentarily", and also because of his Sharingan's resonance with Obito's Sharingan (as both the Sharingan belonged to Obito originally), as is shown in Episode No. 345 of Naruto Shippuuden:





After this scene, Kakashi loses his conciousness owing to the stress put upon him by the activation of Mangekyo. However, Obito goes on to kill the Amegakure Shinobi (Ninja of the Rain village) because of his drive of emotions deeply centred around Rin, and his complete activation of Magekyo.

Thus, you can say that Kakashi had feelings of friendship towards Rin, but not the same as the feelings of love that Obito had.