(Grammer and spelling corrections please refer to Jasmine's entry. )
The way to answer is your own choice. Making yourselves special is the key to get an offer in my opinion.

Q1. Why are you qualified to participate in training program ? explain education, work experience, and preparations you have undertaken.
（I don't understand the meaning of this question...What is qualified...?）

I've been working as a web designer/director for about 2 and a half year.I am able to create websites in Japanese but recently I'm worried about whether my current skills are turning to out-of-date quickly.Technologies used in overseas are the latest, but there's a time lag until they are commonly used in Japan.So I would like to participate in the program to catch up the latest web-creating technologies, to keep on doing this job in the future.

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"Why are you qualified to participate in training program ?" is approximately having the meaning of "Why should we allow you to join this training program?"


This question asked 4 things:
1. The reason of qualifying you;
2. Your education;
3. Your work experience; AND
4. Your preparation

However, you have just explained your work experience and the whole paragraph is only like answering about "why do you want to join?", which is not quite answering the question.
I think that they are all very crucial points to qualify you. Don't miss them.



Q2. What skills and knowledge do you expect to gain during each of the phases of your training ?
(What does "each of the phases of your training" mean...?)

I expect to gain my knowledge of latest web creating technologies such as HTML5,CSS3, and Mobile-first concept.
In addition, since I've created web sites which are full of informations and using many images to look more decorated,I want to gain my experiences of creating text based web sites which inpormations are collected compactly and effectively.

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Phases denotes time spans of the training here. If you know how the training will be undergoing exactly, you may answer it concretely; otherwise, it simply meanings "during the training".




Q3. What’s your purpose in applying for your intern position?　(What is "intern position"...)

In the future I'd like to work with people from various countries to do borderless jobs.I've been working only with people from Japan, so I'd like to take part in the internship program to get used to work with people from various countries.

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Generally, Intern position or internship is a position in a company that the person in the position is mainly learning or practising instead of doing very serious production.
However, the concrete responsibilities of such position is very environment dependent. Since that is the position you are applying, the company should have provided enough information for this internship; otherwise, that would be the thing you can ask the company.



Q4. What are your main strengths?(Nothing...無いってば～ ・∀・;)

I'm a hard worker.I make efforts to do the tasks in a short time,but never cut corners on the job by reason of being too much trouble. I always do my best at any time.

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You are HARD WORKING for SURE!!!!! (good learning ability and activeness would be probably weighted more, to see if you want to add them too.)




Q5. How would successful completion of training program assist you in your future?

If I learn the process of creating websites which are visited from both domestic and overseas through the program,I think I'll be able to adapt to various type of web-creating project in the future.

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I think I'm writing the same things over and over again...(ーー;)

Tip: In a resume be positive.

Q1. [Qualified means you have skills. For example a carpenter's qualifications are wood carving and furniture making ext. A cook's qualifications include cooking and managing the kitchen.] 
I've been have over two years of experience as a professional web designer/director. I am proficient in creating websites in Japanese, and work to keep my current skills up to date in this fast paced felid.

Q2. [Training is any education they provide].
I expect to increase my knowledge of latest web technologies such as HTML5,CSS3, and Mobile-first concept. In addition, since I've focused on creating web sites which are full of informations and images (with a stylish appearance), I want to gain experiences creating text based web sites which information that is collected compactly and effectively to become more well rounded as a designer. 

Q3. [Inters are studying/students and may not get payed, sometimes they get hired into a company after the internship]
I would like to work with people from various countries to do international jobs. I've only been working with other people from Japan, so I'd like to take part in the internship program to gain experience working with dynamic people from various countries. 

Q4. I'm a hard worker. I make efforts to do the tasks in a short time, but never cut corners on the job by due to laziness. I always do my best. 

Q5. If I learn the process of creating international websites through the program, I I'll be able to readily adapt to more varied styles of web-design projects in the future to keep up with this quickly changing field. 

Here we go!!! Lots to talk about. What you have is really good, you just need some refining :D 

Q1. Why are you qualified to participate in training program ? explain education, work experience, and preparations you have undertaken.

What skills/education/work experience/other preperations do you have that will make you successful in this training program? (qualified means "able to do a certain thing")

I've been working as a web designer and director for about two and a half years. I am able to create websites in Japanese, but recently I'm worried about whether that my current skills are turning to becoming out-of-date quickly. Technologies used in overseas are the latest, but there is a time lag until they are commonly used in Japan. So, I would like to participate in the program to catch up with the latest web-creating technologies and to keep on doing this job in the future.

You should also add something about previous training or education in this field.

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Q2. What skills and knowledge do you expect to gain during each of the phases of your training ?
(each of the phases of your training means "each section of training" or simply "while you are training")

I expect to gain my more knowledge on the latest web creating technologies such as HTML5, CSS3, and Mobile-first concept.
In addition, since I have created web sites full of information and used many images to make the websites look more decorated, I want to gain my experiences of experience with creating text based websites where information is collected compactly and effectively (a good word to use here would be "efficiently").

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Q3. What’s your purpose in applying for your intern position?

intern position means a job where you work and learn at the same time (depending on the employers, you may also get paid) I would assume this is the position you are applying for

In the future I'd like to work with people from various countries to do borderless jobs. I've been working only with people from Japan, so I'd like to take part in the internship program to get used to working with people from various countries.

(It would be a good idea to add what you want to learn from the program - it might be a little redundant, but it shows consistency)

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Q4. What are your main strengths?(Nothing...無いってば～ ・∀・;) 間違うよ！

I am a hard worker. I make efforts to do tasks in a amount short time, but I never cut corners on the job by reason of being too much trouble not doing the work correctly. (your phrase doesn't make much sense, but I'm not sure what I've supplied it what you want either, let me know ^^) I always do my best at any time. 

(You can also say that you are a fast learner, deep thinker, etc....)

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Q5. How would successful completion of training program assist you in your future?

If I learn the process of creating websites which are visited from both domestic and overseas through this program, I think I will be able to adapt to various types of web-creating projects in the future.

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I think I'm writing the same things over and over again...(ーー;) Yeah that's typically how text interviews go, at least all the ones I've done hahaha. You are definitely on the right track, though! 

A few grammatical things you should keep in mind when writing formal things...

1. do not use contractions (Write I will instead of I'll, or do not instead of don't)

2. remember to put a space after every punctuation you use (sometimes you forgot to put a space after the period at the end of sentences, or after commas.)

3. double check whether nouns are countable or not (information is a noncountable noun, so that means that you don't need to put an 's' after it to make it repesent more than one information. other noncountable nouns: money, water, rice, etc...)

That's it! Let me know if you have any questions :D 

がんばってください！応援してるよ！