Praise be to Allah.
First of all it should be understood that there is a difference between zakat al-fitr which is given at the end of Ramadan, and zakaah on one’s wealth. Zakat al-fitr is obligatory for every Muslim, to be paid on his own behalf and on behalf of those on whose maintenance he is obliged to spend, if he has wealth surplus to his and his dependents’ needs for the day and night of Eid: one saa‘.
For zakat al-fitr to be obligatory, there is no stipulation of a specific minimum threshold of wealth or that one full year should have passed since acquiring it, or any of the other stipulations that are applicable to zakaah on one’s wealth.
It also has nothing to do with what a person owns of wealth, property or cars, because he pays zakat al-fitr on his own behalf and on behalf of the people on whom he is obliged to spend.
As that is the case, according to what you mentioned in your question, you have to pay zakat al-fitr, on your own behalf and on behalf of your wife and child and your father, if he does not have sufficient wealth to make him independent of means, as you stated in your question.
With regard to the foetus, no zakaah is required on his or her behalf, according to scholarly consensus, but if you do pay zakaah on his or her behalf, there is nothing wrong with that.
What is required in the case of zakat al-fitr is to give it in the form of the usual local staple food.
Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allah have mercy on him) said:
In as-Saheehayn it is narrated that Abu Sa‘eed al-Khudri (may Allah be pleased with him) said: We used to give it at the time of the Prophet (blessings and peace of Allah be upon him), a saa‘ of foodstuff, or a saa‘ of dates, or a saa‘ of barley, or a saa‘ of raisins.
A number of scholars interpreted “foodstuff” in this hadeeth as referring to wheat; others are of the view is that what is meant is the foodstuff of a region, no matter what it is, whether it is wheat, corn, pearl millet, and so on. This is the correct view, because zakaah is a means by which the rich help the poor, and the Muslim is not required to help with anything other than the staple food of his country.
This is the view favoured by Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him). It was also favoured by Shaykh Ibn ‘Uthaymeen and others.
Thus it is clear that zakat al-fitr is to be given in the form of regular food, not cash as mentioned in the question, and not in the form of any other alternative to cash.
Moreover the one who gives the zakaah does not have the right to decide about the way he gives his zakaah, whether it is zakat al-fitr or zakaah on wealth, and buy something else for the poor instead of giving them their zakaah, such as buying meat or clothing and the like for them.
There is nothing wrong with you transferring the zakaah on your wealth, or your zakat al-fitr, to your homeland and giving it to your family there, if they are in need of it. This is confirmed in the case of many workers who work in a country where most of the people are well off and independent of means, when the people in their homelands are in need or are poor, especially when many of them know more about the poor in their homelands than about who is entitled to zakaah in the country where they are working.
This is also confirmed if the zakaah will be transferred from the country where he is working and given to poor relatives of his in his homeland.
Zakat al-fitr becomes obligatory when the sun sets on the last day of Ramadan, and must be paid before the Eid prayer. It is permissible to pay it two or three days before that, if need be.
Based on that, it is not permissible for you to give it a week or two before Eid, and the like.
But if you are afraid that two or three days before Eid is not enough time for the money transfer to reach your homeland, then you can send it earlier, so as to allow enough time, even if that is before Ramadan, and you may appoint someone trustworthy to buy your zakat al-fitr for you, but he should not give it except at the prescribed time.
With regard to zakaah on wealth, as stated above, it has nothing to do with Ramadan or any other month; rather when the wealth reaches the nisaab (minimum threshold) and one full hijri year has passed, it becomes obligatory to give zakaah on it.
If there is a short time left of the year, a month or more or less, and a person wants to hasten his zakaah, it is permissible for him to pay the zakaah on his wealth ahead of time, if there is a need to do so.
For more details on that, please see the answer to question no. 98528
The difference between zakat al-fitr and zakaah on one’s wealth in this regard has been explained in the answer to question no. 145558
For zakaah to be obligatory on cash, two conditions are stipulated:
1.That it has reached the minimum threshold
2.That one full hijri year has passed since it reached that threshold.
If the wealth falls short of that threshold, no zakaah is due on it.
If it reaches the threshold and one full lunar (hijri) year passes from the time when it reached the threshold, zakaah becomes obligatory at that time.
The nisaab or threshold is whatever is equal to 85 grams of gold or 595 grams of silver.
The rate that must be given for zakaah is one quarter of one tenth (2.5%).
With regard to your car which is for personal use, and also the house in which you live, there is no zakaah on either of them.
See the answer to question no. 146692
There is nothing wrong with your father registering whatever he wants of his property in your name, unless he has children other than you, in which case it is not permissible for him to give something to you but not to them. Rather he must treat you all equitably in terms of giving gifts. But if your other siblings agreed to what your father wants to register in your name, without being pressured or embarrassed into agreeing, or being compelled to do so, it is permissible for him to register in your name so long as they are happy with it.
And Allah knows best.