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In your first sentence, "good" may not be the best choice. (For more information, look up the adverbial use of "good".) I would say "better spoken English," "proper spoken English," or just plain "spoken English."

In the third sentence, you cannot regret yourself. You can write "I regret that I have not been able to ..." In speech, you might suppress the "that" and say "I regret I have ..." -- but the suppression is not advised in formal writing.

In the third sentence, you again use "good" improperly. The proper phrase is "who know English well."

In the fourth sentence, you want to add the conjunction "and" between "way" and "without" -- "in an effective way and without mistakes."

In the last sentence, "also" is misplaced. It should go after "I" -- "I also want to improve my accent."

(Don't way about your accent. Concentrate first on fluency -- being able to speak while "thinking on your feet". If you worry about both at first, you will not be able to improve either.)

(1) Your use of the personal pronoun "I" is not a problem in this type of writing. Normally in formal writing, we try to avoid it. We don't say "I believe" in an essay because it is considered redundant -- as the author, you are already expressing what you believe. But in something this personal, you cannot avoid "I".

(2) Yes, you can say "English speaking skills". In this case, you can even properly say "good English speaking skills."

(3) See my original answer regarding the use of good. If you properly use the word, you will reduce the number of times it appears.

(4) You can say "I know a little grammar," but "I know grammar a little" is appropriate only in an informal context. There is also a difference in meaning. Somebody who knows "grammar to some extent" probably knows more than somebody who knows "a little grammar".

(5) You might say "I struggle to construct sentences ...." or "It is hard for me to construct sentences ..."

(6) You can say "to construct sentences effectively". Note that "to construct effective sentences" might also be appropriate here, but it carries a different meaning.

I hope this also helps.

Sanjay, I don't have any additional suggestions on the grammar of this passage so I will comment on the content.  Even those of us native speakers who are grammar geeks or wordsmiths write more properly than we speak.  If all of the words I spoke in a day were written down, you would find all sorts grammatical errors and ill-chosen words.  What you can't write down is the tone of voice, the "uh" & "errrr" moments, the shrugs or faces that I might make.  Non-verbal communication helps fill in the blanks. The other person in the conversation can infer meaning or even fill in a word for me if the right one doesn't spill out.  Sometimes (now that I think about it) I use other people as a thesaurus when trying to find the right adjective.  I know I have a word that is close to what I mean, but not quite, so they suggest other words until we hit on the right one.

You gave someone else some great advice the other day - it takes practice.  If someone offers a better word while you are talking, don't feel bad.  It is completely normal.  Improving your accent will likely be the most difficult.  I know that when people move to other countries they tend to lose their accent over time, but they are completely immersed in a new environment.