I wish to ask a few questions on the half shekel
- To whom is it given today?
- Is the amount really significant nowadays?
- If so, does it differ with different currency values or do all Jews pay the same?
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- The money is given to charity in memory of the half-shekel donated annually to the Temple treasury in the month of Adar. The money should be given to the poor. It should not be used for synagogue upkeep, communal needs, or a similar cause.
- The custom is to give three times this half unit (e.g., $1.50) because parshas Ki Sisa uses the word “terumah” (donation) three times
- It is questionable whether a person may use coins that are not legal tender in his country, and he should rather give [three half-coins of his own currency, if not] three whole coins and have in mind that half of each coin is to fulfill the mitzvah, and the other halves are a gift to charity. Alternatively, two people could jointly give three whole coins. In extenuating circumstances, one can fulfill this mitzvah by giving any sum of money to charity.
See also here from Eretz Hemda.
To add to @mbloch's answer regarding your first question, there was an old custom in Ashkenaz to give machatzit hashekel either to those moving to Israel, or to support the poor living there.
וראיתי שם שבעלי כיסים ויש מן הבינונים נתנו מחצית השקל, ואת מחצית השקל החזיקו לסייע בו אל העולים לארץ הצבי לשם שמים.
I saw there that the wealthy people, and also some people of moderate means, gave machatzit hashekel, and they would use the machatzit hashekel to support those ascending to the Land of Israel for the sake of Heaven.
בהרבה מקומות המנהג לשלוח מחצית השקל לעניי ארץ ישראל, ומנהג יפה הוא, אחר שבזמן הבית היו כל ישראל שולחין שקליהן לירושלים.
In many places the custom is to send the machatzit hashekel to the poor of the land of Israel, and it is a beautiful custom, since in the time of the beit hamikdash all Israel sent their shekalim to Jerusalem.