1. What should be the intention for visiting the Prophet's mosque?

Is it permissible to have the intention of ‘ziyarah’ of Prophet’s ‘raudah-e-mubarak’ (grave)?

2. Is it permissible to pray in the raudah?

3. Is saying salaam and ‘khadam bosi’ to Prophet (touching one’s feet out of respect. In India, on certain occasions like Eid or leaving on a journey, people touch the feet of their elders.

This act, they say, is based upon the hadith that ‘Paradise is beneath the feet of the mother’) permissible?

4. What are the acts of Sunnah to be performed when visiting Madinah Munnawwara?

5. Is this hadith regarding visiting Masjid Quba authentic?

"Anyone who came for ziyarah of my masjid and did not visit Masjid Quba, had made an act of atrocity"

1. The intention should be to visit the Prophet's (ﷺ) mosque and not the grave. It is not allowed to have an intention to visit the grave.
2. Praying in the raudah روضة is permissible if the authorities allow.
3. The hadith that ‘Paradise is beneath the feet of the mother’ means serving her and obeying her. It is allowed to touch the parents' feet or kiss them but this can be done at any occasion and not specifically when one is going on a journey or on Eid. This should not involve bowing because bowing has connotations of worship and worship is only for Allah. The idea behind kissing their feet is to show love and respect and to believe that by kissing or touching, you take blessings from the feet is bid'ah. Saying the ‘khadam bosi to the Prophet (ﷺ) is bid'ah. The Sunnah way is that the person goes and makes salaam for him and not on behalf of others. Whoever is not in Medina can make salaam and Salat upon the Prophet (ﷺ) wherever he is and Allah will hear it and know that.
4. The Sunnah for visiting Medina is to perform the compulsory Salah’s in the masjid and also the nafil because the reward for prayer there is multiplied. That is what a person gains most by a visit there. A visit to Masjid Quba can also be made because the Prophet (ﷺ) used to visit this masjid and pray there weekly.
5. I have not heard the hadith that you mentioned concerning Masjid Quba. The popular and well-known evidence for visiting the Mosque of Quba is the hadith narrated in al-Saheehayn from Ibn Umar (radiAllahu anhu) who said: “The Prophet (ﷺ) used to come to Quba riding and walking.” According to another report: “and he would pray 2 rakats there.” [Al-Bukhaari and Muslim]. The Prophet (ﷺ) also said, “Whoever purifies himself in his house then comes to the mosque of Quba and prays there, he will have a reward like that for ‘Umrah. [Ahmad, al-Nasa’i, Ibn Maajah and al-Hakim. Al-Hakim classed it as Sahih and al-Dhabi agreed with him. Also classed as Sahih by al-Albani in Sahih al-Jami’, 6154]