Giving a gift to a non-Muslim
The way gifts can be exchanged between Muslims, in the same way, Islam permits exchange of gifts between Muslims and non-Muslims as well. There is a Hadith in Jame’ Tirmidhi about this:
عَنْ عَلِىٍّ عَنِ النَّبِىِّ -صلى الله عليه وسلم أَنَّ كِسْرَى أَهْدَى إِلَيْهِ فَقَبِلَ مِنْهُ وَأَنَّ الْمُلُوكَ أَهْدَوْا إِلَيْهِ فَقَبِلَ مِنْهُمْ.
Translation: It has been narrated on the authority of Hadhrat ‘Ali (May Allah be well pleased with him) that the emperor of Iran sent a gift to the Holy Prophet (Sallallahu alaihi wa sallam) and He accepted it. Similarly other kings also sent their gifts. The Holy Prophet (Sallallahu alaihi wa sallam) accepted them.
There is another similar Hadith in Jame’ Tirmidhi:
أَنَّهُ كَانَ يَقْبَلُ مِنَ الْمُشْرِكِينَ هَدَايَاهُمْ
Translation: It has been narrated that the Holy Prophet (Sallallahu alaihi wa sallam) accepted gifts from non-Muslims.
Similarly accepting invitations of non-Muslims and inviting them to Muslims’ functions, etc. is also permitted in the Shariah, provided there is nothing there which goes against the Shariah. As mentioned in this Hadith of Musnad Imam Ahmed bin Hambal:
أَنَّ يَهُودِيًّا دَعَا رَسُولَ اللَّهِ -صلى الله عليه وسلم- إِلَى خُبْزِ شَعِيرٍ وَإِهَالَةٍ سَنِخَةٍ فَأَجَابَهُ
Translation: A Jew invited the Holy Prophet (Sallallahu alaihi wa sallam) to a "feast" of barley bread and stinking fat and the Holy Prophet (Sallallahu alaihi wa sallam) accepted it.
It is given in the Holy Quran in this regard:
وَطَعَامُ الَّذِينَ أُوتُواْ الْكِتَابَ حِلٌّ لَّكُمْ وَطَعَامُكُمْ حِلٌّ لَّهُمْ
And the sacrificed animal of those given the (Revealed) Book is (also) lawful for you, whilst your sacrificed animal is lawful for them.
However, animals sacrificed in the name of anyone else except Allah are forbidden.