Firstly: 

This contract is a form of hire-purchase, concerning which a statement has been issued by the Islamic Fiqh Council, stating that it is permissible if the rental contract is accompanied by a separate contract giving the item to the renter so long as he pays off all the rent, or a promise to give it to him after he has paid all the rent. End quote. 

See the text of the statement in full in Fiqh al-Nawaazil by Dr. Muhammad Hasan al-Jeezaani (3/301). 

Giving something for free subject to a condition has been ruled to be valid by some scholars. This is the view favoured by Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him). But that is subject to the condition that the contract of giving is separate from the rental contract, which states that the item is to be rented for a certain amount of money, for a certain length of time, then states that it is to be given subject to certain conditions,  such as saying: The two parties agree that the first party (the company for example) will give the second party (the customer) the item for free, if he pays off the instalments that he owes. 

It is also essential that the rental be a real rental and not just a cover for selling. So the one who rents it out is liable for the rented item, not the one who is paying rental for it.  Similarly the costs of basic maintenance other than running costs are to be borne by the one who rents it out, not the one who is paying rental for it, for the duration of the rental period. This is different from a sale, when the liability and mantenance are all borne by the purchaser because he takes possession of the item as soon as the contract is completed. 

Secondly: 

What you say about “If payment is made before the first day of every month, it will be considered paid for 3 months, while if you pay after the first day of the month then it will be considered paid for only one month” is known to the fuqaha’ as “waiver with early payment”, i.e., if the borrower hastens to pay what he owes, some of the debt will be waived for him. 

If you hasten to pay off the debt ahead of time, two instalments of the amount you owe will be waived for you. 

The fuqaha’ differed as to whether this is valid or not. Most of them are of the view that it is not permissible, because it is akin to riba. Ahmad and al-Shaafa’i, according to one of their two views, were of the view that it is permissible, and this is the most correct view. For a detailed discussion on that, please see the answer to question no. 13945. 

In a statement issued by the Islamic Fiqh Council it says: Waiving part of the debt that is not due yet, whether it is at the request of the lender or the borrower, which comes under the heading of “waiver with early payment”, is permissible according to sharee’ah, and it does not come under the heading of forbidden riba so long as it is not based on prior agreement. End quote from Majallat Majma’ al-Fiqh (2/217). 

Hence it is known that this form of waiver with early payment is not permissible, because there was prior agreement to that at the time when the contract was drawn up. 

And Allaah knows best.