Is it permissible for a person, who vowed to offer two Rak`ahs (units of prayer) everyday for Allah`s sake, to do so after the Fajr (dawn) prayer?

In principle, a vow should be fulfilled as it was originally intended, and if its time lapses then, it should be fulfilled at any time because of the vow. Therefore, it is permissible to offer them after the Fajr prayer, and there is no prohibition in doing so. However, it is preferable for him to offer these two Rak`ahs at some other time to avoid the disagreement amongst the scholars in this regard.

Should one who doesn`t perform prayer out of laziness make it up later, and how should he/she do so?

All perfect praise be to Allah,The Lord of The Worlds                                                                                                                                                              He/she is obliged to make up missed prayers by offering with each obligatory prayer another one, and if he/she offers two, then it is better. And Allah Knows Best.

What is the ruling on a Muslim who slaughters an animal while being in a state of Janabah (ritual impurity)?

All Perfect Praise be to Allah, The Lord of The Worlds, and may His Peace and Blessings be upon our Prophet Muhammad and upo all of his family and companions.

Slaughtering is permissible for the ritually impure man and the menstruating woman because ritual purity isn`t a condition for the validity of the slaughtering. Moreover, it is permissible for the uncircumcised person to do the slaughtering as well. And Allah The Almighy Knows Best.

Should the Zakah (obligatory charity) giver tell the poor recipient that this is the Zakah of his money?

No, he shouldn`t. But, the Zakah giver should make an intention in his heart that it is the Zakah of his money when paying it to the poor. And Allah Knows Best.