suggest

1a. She suggested his/him applying for a work permit.
1b. She suggested him to apply for a work permit.
2. What you suggest to improve my English language?
3a. Could you suggest to me a substitute medicine?
3b. Could you suggest a substitute medicine to me?
Please correct all.
Thanks.

I suggest that you take a look at all the questions you’ve asked in the past where you’ve tried to use ‘suggested/suggest’.
You are still making the same mistakes.

1a. She suggested that he (should) apply for a work permit.
1b. She suggested to him that he should apply for a work permit.
2. What do you suggest to improve my English language?
3a. Could you suggest a substitute medicine for me?
3b. Could you suggest a substitute medicine to me?

Beeesneees,
1.
For ‘What do you suggest to improve my English language?’
can I write as below:
‘What do you suggest for improving my English language?’
Is it OK?
2. I suggest you to be there at 5.
Sometime before I think you told this sentence is wrong.
Am I correct?
Thanks.

  1. ‘To improve’ is definitely the better option there.

  2. Yes, I told you that was wrong. It needs to be:
    I suggest that you be there at five.

Are the words I have highlighted in the quote acceptable/necessary?

Perfectly acceptable, but not necessary. However, they are required if you wish to make it clear that the suggestion was made directly to him. If you omit them, thew it is possible that the suggestion was indirect - or even just a general observation.

I believe that in this sentence -

‘to him that he should’ is unnecessarily longwinded. There is surely no ambiguity in writing -

She suggested he apply for a work permit.

Though I tend to agree more with you for its brevity, a new learner is likely to wonder how ‘he apply’ is used when the subject is singular. So, at least at one place if it is shown in full [She suggested (to him) that he should apply for a work permit], the learner might understand it the following way and be convinced that ‘should’ is implied/unexpressed:

She suggested to him that he should apply for a work permit.
She suggested that he should apply for a work permit.
She suggested he should apply for a work permit.
She suggested he apply for a work permit.

Fine, let’s remove the present subjunctive and use the past subjunctive ‘applied’ or lose the subjunctive altogether and make it positive ‘applies’. Either way is better than the protracted version.

Do you mean to say the following are also acceptable?

  1. She suggested he applied for a work permit.
  2. She suggested he applies for a work permit.

If so, I have no comments.