I agree with Leigh Mumford's corrections for the most part.

The baby hippopotamus bit me on the right calf.  How dangerous the animal is!  It didn't stop bleeding so I sprinkled styptic powder over the wound.
I got in touch with my mother that night.
"How are you getting on with the hippopotamuses?" she asked.
"Mom, they have a stick up their ass," I replied.  "I'm through being their garbage boy.  I mean I've quit my job as a zookeeper. By the way, when will you be here?"
"I'll arrive there at approximately 8:10 A.M.  See you in a bit."
"Yes, mommy."

I would move the text the way I showed.  If you are writing out a conversation it is best to designate speech by the same person to one line or paragraph, generally speaking.

"How are you getting on with the hippopotamuses?" said she.
"Mom, they have a stick up their ass." replied I.
"I'm through being their garbage boy. I mean I've quit the job of zookeeper. By the way when will you be here?"  <em><- If you place it on the following line, it looks like this is a reply to what you've said instead of something you said yourself.</em>
"I'll arrive there at approximately 8:10 A.M.  See you in a bit."
"Yes, mommy."

When quoting speech the punctuation with sentences that end in periods can be confusing: 

<em>"My name is Billy," said Billy.</em>

<em>"My name is Billy!" exclaimed Billy.</em>

<em>"My name is Billy?" asked Billy.</em>

You can simply end the sentence as shown and start a new one:

<em>"Mom, they have a stick up their ass."  I wondered what she would think about that.</em>

Or the following is correct:

"Mom, they have a stick up their ass." I replied.

"Mom, they have a stick up their ass," I replied.

I don't think you need to include "I mean..." in the beginning of the sentence since the context clues make it self explanatory, but it's not grammatically incorrect to do so.  You can also connect the two sentences with a semicolon:

I'm through being their garbage boy; I've quit my job as a zookeeper.

I'm through being their garbage boy!  I've quit my job as a zookeeper.

When you say phrases like "on that note" or "by the way" or "speaking of which" they should be followed by a comma.

<em>By the way, what are you doing for lunch today?</em>

<em>Speaking of which, how's your brother doing?</em>

<em>On that note, I just wanted to let you know how much I appreciate you.</em>

<em>
</em>

Regarding the phrase, "They have a stick up their ass," it implies that all the hippopotamuses share one ass.  "They [all] have sticks up their asses," is more grammatically correct.

These statements are okay because "they" share the one thing that "they" own:

<em>They have one dog.</em>

<em>They have a beautiful home.</em>

<em>They have a black car.</em>

These statements are not okay:

They have a lot of hatred in their heart.  [They do not all share one heart.]

They have a lot of crazy ideas in their head.  [They do not all share one head.]

Practice composition

I made wrote this text. Is there anything to correct? Please correct it if any.

The baby hippopotamus bit me on the right calf. How dangerous the animal is!
It didn't stop bleeding so I sprinkled styptic powder over the wound.
I got in touch with my mother that night.
"How are you getting on with the hippopotamuses?" she asked said she.
"Mom, they have a stick up their ass." I replied I.
"I'm through being their garbage boy, I mean I've quit the my job of as a zookeeper. By the way when will you be here?"
"I'll arrive be there at approximately 8:10 A.M. See you in a bit."
"Yes, mommy."

See you in a bit... it is you usually being used when you want to say goodbye

It's bettet to say "see you soon" or "see you in few minutes/hours"

I guess there is no other thing!