Truncating the person doing the action in English
Since I've been learning Spanish, I've become aware of what is probably bad English. It's very common to hear or say any of the following (as well as other things), but is it correct? Because this is commonly done, would you teach a Spanish speaker this "exception"?
Hope you feel better. (I hope you feel better)
See you soon. (We will see you soon)
Love you. (I love you)
I suspect this happens in every language.
Your example of "Hope you feel better soon" reminds me of "Espero que tengas un buen dia" where the "Espero" is dropped and most people just say "Que tengas un buen dia."
I think it depends on what the teaching situation is.
If it's an informal group of people who would have logical occasion to use some of the examples you cited, then it would make sense to bring these things to their attention, especially if learning English comes relatively easily to them and if it doesn't seem that it would confuse the issue. (Hey, if you're sitting around drinking mate with a bunch of friends and smearing dulce de leche on galletas, go for it!)
On the other hand, if you are teaching English that your students will need to use on the workplace, it might be better to hold off on some of the shortcuts we use in English until they have a solid grasp of the standard structures.
....esos son algunos pensamientos de una vieja profesora...
It depends, I guess if you are a proper speaker - for example, during an interview or something like that, you say everything correctly to make a good impression. Whereas in every day conversing you don't really focus on the parts that aren't so needed. So you would say the 'I' etc. In conversing you may miss it out, like you've pointed out in I love you, you will just say Love you.