Is it permissible to pray for forgiveness and mercy for a kaafir who is still alive?

Firstly: 

We ask Allah, may He be exalted, to guide your mother to that
which He loves and which pleases Him, and to make her one of the devout
women who worship Allah, and to grant you the joy of seeing her guided and
becoming righteous. And we ask Him to unite your family in obedience to Him
in this world and in the highest degrees of Paradise in the Hereafter. 

Secondly: 

What the seeker of knowledge told you about it being
permissible to pray for mercy for a non-Muslim while he is still alive is
correct in the sense that Allah may bestow mercy upon him by saving him from
sin and guiding him to obedience, and thus transferring him from being one
of those who are subject to His wrath to being one of those upon whom He
bestows His mercy. 

The more correct view, concerning the correctness of which
the scholars are unanimously agreed, is that supplication (du‘aa’) may be
made for guidance for a kaafir who is still alive, but not for mercy or
forgiveness. By doing so, one will avoid an area of scholarly differences
concerning this issue. 

Among the evidence that this is more correct is the report
narrated by al-Tirmidhi (2739) and Abu Dawood (5038) from Abu Moosa (may
Allah be pleased with him) who said: The Jews would pretend to sneeze in the
presence of the Prophet (blessings and peace of Allah be upon him) hoping
that he would say to them, “May Allah have mercy on you,” but he used to
say: “May Allah guide you and set your affairs straight.” 

See the answer to question no.
43164. 

Thirdly: 

There is no difference of opinion among the scholars that one
cannot pray for forgiveness and mercy for a kaafir who died in a state of
kufr. 

Al-Nawawi (may Allah have mercy on him) said: 

Offering the funeral prayer for a kaafir and praying for
forgiveness for him is haraam according to the text of the Qur’aan and
scholarly consensus. 

Al-Majmoo‘, 5/119. 

Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him)
said: 

Praying for forgiveness for a kaafir is not permissible
according to the Qur’aan, the Sunnah and scholarly consensus. 

Majmoo‘ al-Fataawa, 12/489 

With regard to praying for mercy and forgiveness for a kaafir
who is still alive, there are numerous comments of the scholars concerning
the permissibility of doing so, but not in the sense of asking for
forgiveness for his shirk and kufr if he dies believing in that, or asking
for mercy for him from his Lord when he has met Him as a kaafir. Rather,
such supplications are to be understood as seeking to achieve that which may
be a cause of his being forgiven and shown mercy, which is that he be guided
to Islam. This is one of the ways in which we may interpret the words of
Ibraaheem (peace be upon him) as mentioned in the verse (interpretation of
the meaning): “O my Lord! They have indeed led
astray many among mankind. But whoso follows me, he verily is of me. And
whoso disobeys me, –
still You are indeed Oft-Forgiving, Most Merciful” [Ibraaheem 14:36],
and the words of the Prophet (blessings and peace of Allah be upon him): “O
Allah, forgive my people, for they do not know.” Narrated by al-Bukhaari
(3290) and Muslim (1792). 

Ibn al-Qayyim (may Allah have mercy on him) said:

“And whoso disobeys me,
– still You are indeed Oft-Forgiving, Most
Merciful” – it does not say “still You are indeed Almighty, Most
Wise”, because the context speaks of seeking the kindness of Allah and of
du‘aa’ (supplication) i.e., if You forgive them and have mercy on them by
enabling them to recant shirk and accept Tawheed, to give up sin and turn to
obedience, as it says in the hadeeth, “O Allah, forgive my people, for they
do not know.” 

Madaarij al-Saalikeen, 1/36, 37 

Badr al-Deen al-‘Ayni (may Allah have mercy on him) said,
commenting on the hadeeth, “O Allah, forgive my people, for they do not
know”: 

What this means is: guide them to Islam with which
forgiveness becomes valid, because the sin of kufr cannot be forgiven. Or it
may be that what is meant is: Forgive them if they become Muslim. 

‘Umdat al-Qaari Sharh Saheeh al-Bukhaari,
23/19 

al-Haafiz Ibn Hajar mentioned these two interpretations in
Fath al-Baari, 11/196 

To sum up: 

It is better to pray that your mother be guided and that
Allah open her heart to doing what He has commanded her to do. But if you
pray for mercy for her, there is nothing wrong with that, with the intention
that Allah may bless her with guidance to do that which earns mercy, i.e.,
prayer. 

To learn about the best way to call one who does not pray,
see the answer to question no.
47425. 

And Allah knows best.