Informal letter

Dear Henry,
Hi, how are you doing? I hope you're well. It's been a while i haven't sent you letters (It's a been a while since I've written you a letter). You may be asking why (It would sound better to say "You might be wondering why."). It's because was not something I had nothing interesting to tell you about. But in this letter, I am going to tell you about my first time attending going to the zoo.
This past Saturday I accumulated some money and decided to spend it in at the zoo. I also invited a few friends to get have/make it more fun. So we bought tickets and started to seek (more natural to say "look for" or "check out") the animals that interested us. The first one that we found was an alligator. Observing this dangerous animal we discovered for ourselves a few new facts about Its behavior. The next animal was the flamingo. This weird pink bird made a great impression on us.Then we saw a few more weird and interesting animals. But the most impressive one was an African elephant. The enormous beast scared us at first but appeared that it was a pretty quiet animal.
I really had a lot of fun during this visit. I wish you were there. I would like to invite you the next time.
All the best, Kolya

In the first two corrections above I kept two sentences as you originally had. However, they can be combined as I've done below.

"I know that I haven't written you in a while and you've probably been wondering why." 

Other than that the letter was fine grammar-wise. Most of my corrections were just spelling mistakes. Good job! 

You should start to do the dishes. The verb "start" can be followed by a to-infinitive or gerund with no change in meaning. You should start doing the dishes.

Today

Well, I felt really tired today. I went to my German class in the morning. What did I learn? I learned how to give directions to people and how to say the time. When my class finished, I went to the bus station. There was were a lot of people there! The good thing was that I had someone to talk to/with, so the time felt went by fast. Suddenly I was in at home. and I started to do all of my homework, waiting to finish someday :c but waited to finish it later.

"Suddenly I was home." This would sound more natural as "Before I knew it I was home."

I rephrased the last sentence to better express what I think you were trying to say. My apology if I was not correct in understanding you. 

It's a fancy way of saying "please indulge me in what I'm saying". It can have a few meanings but in this case, it's the speaker acknowledging that they're unsure what to call their methods. So they're saying if you will just indulge them in saying techniques or skills. You might see this when someone is asking you to imagine something specific. It's a little dramatic. "Imagine, if you will, a glorious sunset..." In other words, indulge me in picturing this scene in your mind.

1. It would sound much better as "obstacles". 2. There is no contradiction. It's implied that all students have the opportunity to get the award.

With context, the meanings can vary a little. They both refer to a previous location. "Go back" suggests that the speaker is telling you to go back to a previous location. This location would be away from the speaker. So, leaving the speaker. "Come back" suggests the speaker is also asking you to return somewhere. However, this location would be where the speaker is. So, returning to the speaker. Examples: Person A: I forgot my cell phone at home. Person B: Go back and get it. So here "go back" implies person B is not going to go back with person A. So person A is returning to a previous location alone. Person A: Come back home after school. Person B: Ok, I will. Here person A is asking person B to return to a previous place where they still are. So, they will then be in the same place.

I am listening to the radio and I hear a girl singing on the radio. This use of listen and hear in this sentence is correct. However, the sentence could be phrased more naturally as "I am listening to the radio and I hear a girl singing." You could even split them into different sentences. "I am listening to the radio. I hear a girl singing on the radio." Is this sentence correct? Hope that helps!