Here are some thoughts on the first essay:
Fathers' always have words of wisdom for their children.
No need for the apostrophe there. It is the plural of father--fathers.
The simple advice from my own father that has shaped me as a person is to "pursue something that you are passionate about, and do not forget to take some risks along the way."
One way to get that word count down is to phrase your verbs in the active voice. Try to get rid of as many "to be" verbs as you can.
I discovered that it takes a lot of hard work to be a sole owner of a business, but enjoyed the challenge of running a business.
Here's another place you could cut back on your word count. Take out "of running a business." It is redundant and doesn't change the meaning of the sentence. Putting this into the active voice would save even more words ... Owning a business requires hard work, but I enjoyed the challenge. From 26 words down to 11.
I had always been interested in technology and business, so I looked there for my passion.
This is wordy without saying a whole lot. This could be boiled down to: I found my passion in technology and business. 16 words to 8. Changing this doesn't make it fit as well where it was, but I wanted to give you ideas for cutting the word count. If you cut with this kind of ruthlessness, you will have room to elaborate on certain points and will actually be saying MORE in your essay.
I began learning everything I could about web design and development, flash animation, and motion graphics.
Make this stronger: I taught myself about web design ... Not only will you save words, but it makes more of a statement. "Everything I could," weakens the sentence because it means different things to different people. Omitting "everything I could" lets the reader assume a greater level of competency with what you have learned. Teaching yourself shows more initiative than merely learning.
I became completely self-taught in all of these subjects, but specifically enjoyed creating websites in my spare time.
This becomes redundant with the changes in the other sentence, but you could work the part about creating websites into the other sentence. Again, I would omit the part about "spare time." You don't need that as a qualifier and it limits the amount of accomplishment in the reader's mind.