Hey, could you please go over the below sentences and tell me if they sound natural? The phrases in uppercase are the ones that I want to memorize so if a sentence doesn't read a good English, could you please provide another example with such a phrase?

1. It seems to me that he will CAST HIS LOT WITH his predecesors.
2. More and more people FEEL THE BRUNT of high inflation.
3. The director asked me to GIVE HIM A RUNDOWN ON the progres of the project we are working on.
4. I CAN’T QUITE PUT MY FINGER ON why the application is not working.
5. He fired her UNBEKNOWNST TO ME.
6. I don’t fancy myself a good cook, I’m afraid I won’t be able to meet your FASTIDIOUS taste.
7. It TOOK ME SOME TEMERITY to ask for a pay rise.
8. I”VE BEEN GIVEN A LEEWAY when it comes to whom I can hire.
9. I try to STRIKE A PERSONAL RAPPORT with the people I work with.
10. We have a PANOPLY OF options that we might consider.

Question

Hey, could you please go over the below sentences and tell me if they sound natural? The phrases in uppercase are the ones that I want to memorize so if a sentence doesn't read a good English, could you please provide another example with such a phrase?

1. It seems to me that he will CAST HIS LOT WITH his predecesors.
2. More and more people FEEL THE BRUNT of high inflation.
3. The director asked me to GIVE HIM A RUNDOWN ON the progres of the project we are working on.
4. I CAN’T QUITE PUT MY FINGER ON why the application is not working.
5. He fired her UNBEKNOWNST TO ME.
6. I don’t fancy myself a good cook, I’m afraid I won’t be able to meet your FASTIDIOUS taste.
7. It TOOK ME SOME TEMERITY to ask for a pay rise.
8. I”VE BEEN GIVEN A LEEWAY when it comes to whom I can hire.
9. I try to STRIKE A PERSONAL RAPPORT with the people I work with.
10. We have a PANOPLY OF options that we might consider.

Answer

Excellent work!  Most of your answers are spot on.  Here are a few corrections:
3. progress  (spelling)
7. Omit ME
8. Omit A.  Leeway is noncountable and requires no article.  However, in British English the article 'the' is used in the phrase, 'make up the leeway'. 
9. I'm not sure if the phrasal verb 'strike up' might be better here.  We usually use the verb 'build' with rapport.

* Many of the vocabulary words in these sentences are very formal and not commonly used.

Answer

Overall it is fine, it is just that these phrases are less commonly used. They are still common in certain situations, so you can use them there (usually more formal ones). Hope this helps! :)