"Kindly let us know" is a very formal (and somewhat British) way of concluding a letter that has requested some information. For example, your company offered someone a job, and the office of personnel sent that person a letter making an offer of employment to them. The formal letter would conclude with something like "Kindly let us know of your interest in the position, so that we may arrange a starting date for your employment." So, yes that phrase does contain proper grammar, but today, most people's writing style tends to be more conversational, so you would more likely see "please let us know."
no, that is definitely not proper grammar.
No
Make sure it follows the rules for proper grammar, contains no spelling errors or any 'alternate spelling', and that it actually makes sense.
The proper grammar for that sentence is "She was much better than yesterday." This sentence is in the past tense and correctly compares her current state to how she was yesterday.
The proper way to say it would be the second one, "You are kindly requested to decide the matter." The first one says that they are requested to decide kindly, which doesn't mean the same thing at all.
The sentence "I wish you were as smart as they." is grammatically correct. It is shortened from "... as smart as they are."
Yes, "The house is not as cheap as he thought" is proper grammar.
Yes. That sentence is proper as written.
Yes, starting a sentence with "then" is grammatically correct, especially when showing a sequence of events or actions. However, it is usually used sparingly to avoid overuse and maintain variety in sentence structure.
The proper grammar is "There has come a time." "There" is the subject of the sentence and is singular, hence the correct verb form to use is "has" instead of "have."
Yes. For example, What did she point at ? However, do not use 'at' with 'where.'
The proper grammar is "a urinary" because the word "urinary" begins with a consonant sound, even though the first letter is a vowel.