Both are correct.

It is not related to a gerund. Remember that in English we use "-ing" for two different purposes. Yes, one purpose is to form a gerund (use a verb as a noun). Another purpose is when speaking in the present continuous tense. We use the verbs in their "present participle"form, which is created by adding "-ing". Eg. I run (present simple), I am running (present continuous). What are you doing right now in this moment? I am running. In "I get off the phone (feeling ashamed)", "feeling ashamed" is an adverbial clause. It describes HOW you performed the action of getting off the phone. If you had to separate the two clauses, you could say: "I get off the phone. I am feeling ashamed." This second clause is written in the present continuous tense because AT THAT MOMENT you are feeling ashamed. That is why the verb is in the "-ing" form: it is a present participle. But, when we combined to two sentences together into the more complex sentence structure, it is not necessary to include "I am" anymore.

Enclosed is the* report that you requested. This document *was supposed to be sent out* with the goods, however, we have sent it to you via email in order for DHL to be able to pick up the goods on time. We hope *this is not a problem for you*.

...rice crackers tend to get* stale easily.

Yesterday I got ticketed by a police officer when I was driving to the workshop. He claimed that I didn't perfectly stop* the car at the blinking red light. I wanted to make a fuss because there was another car that hadn't* perfectly stopped there either*. But he seemed so determined that I couldn't.

How CAN you? How is it POSSIBLE that...? "Could" is used to express possibility. Using the phrase "how could you", you are asking how it is possible for a human being to do such a thing. It is true: could is the past tense verb of can, but we also use it to speak generally in the present tense. The clue to this is in the second half of the sentence where it states that the daughter "can't take care of herself". This is the present tense. Presently and in the immediate future, the daughter can not take care of herself and so abandoning her would be the worst thing for a parent to do. In general, how does someone have it in their heart to do that to their own child? #2 in your attempted answer is correct but it is important to note the use of "could" to take about possibility. The sentence you have given asks how it could be possible for a person to abandon their daughter. It implies that this situation is crazy or immoral and that the speaker can not imagine themselves making the same choice or deciding on the same action (abandoning the daughter).

Yesterday (Please help me correct my horrible writing)

It was at 6 am when I got up yesterday morning. It was so early that my other three roomates were still in bed. Having not gone to the memorial tower for several months, I took a walk to get there. Seeing the green trees and high mountains and breathing fresh air, I felt very pleasant. I believe that going to sleep early and waking up early is good for my health. After walking for a while, I decided to take photos of this beautiful scene to motivate me to get up early in the morning. Finally, I picked two pictures I was satisfied with, to be saved them and deleted the others. At that moment, I felt a little bit hungry and the cold wind made the hunger more intense. So I went back to my dorm, and buying my breakfast on the way. After my breakfast, I started washed my clothes and hung them outside. Next, I went to the playground as the sun was high in the sky. My shoulder hurts because I had been seated myself on the chair for a long time. Maybe I need to do more body exercises. I found there were some professional football players from Africa running on the playground. I sat quietly, watching them run around. They sweated a lot.

Hi guys! I'd like to improve my English. Could you correct my text please??

It is said that sleeping is one of the best things in the world. Although I agree with them, I must say that I just can’t get enough sleep since I suffer from insomnia. Even if I’m sleepy, every time I lie on the bed my heart starts beating very fast and my brain starts over thinking. So I start to counting the sheep but it doesn’t usually work. Then I start thinking about how many hours of sleep I need until the alarm clocks rings and I get nervous because I won’t get enough hours of sleep. The result is that I find myself with big bags under my eyes in the morning.