My question is regarding the letters that begin some surahs. Is it permissible to join those letters with the first ayah of the surah

My question is regarding the letters that begin some surahs.  Is it permissible to join those letters with the first ayah of the surah, and in doing so, do you apply the rules of idgham if they apply? eg Surah Yasin and Surah Al Qalam

In particular, is it permissible to join the letters at the beginning of Surah Al-Imraan with the beginning of the surah, and if it is, how would it be done?

Jazakallah khairan. Wasalaam

Yes, it is permissible to join the individual separate letters which begin some surahs with that which comes next.  The question of whether we apply the rules of idghaam or not depends on which riwaayah or qira'ah we are reading.  In Hafs 'An 'Aasim min tareeq Ash-Shaatibiyyah, we do not apply the idghaam rule when joining the separate letters at the beginning of  surahs Ya Seen and Al-Qalam when we join them with that which follows, and instead read the with  :

.  In Tareeq At-Tayyibah for Hafs, there are some turuq which have the idghaam of the saakinah into the which follows it,  and some other turuq in which  the saakinah is read  with (as in tareeq ash-Shaatibiyyah) when joining with the which follows.

Yes, it is also permissible to join the beginning separate letters of Aali 'Imraan, which are with the next word, which is the Glorious name of Allah .  In this case we will have the meem saakinah of followed by a saakin as the first pronounced letter in the Glorious name of .  We have two allowed ways of reading the invidivual separate letter of when joining the two words in one breath, the first allowed way is the normal six vowel count lengthening of al-medd al-laazim on the while putting a fathah on the ; the second allowed way when joining the the of with the Glorious name of in one breath is to lengthen the only two vowel counts with a fathah on the .  This is because there are two saakin letters meeting between two words which the Arabic language does not allow, so the saakinah must acquire a vowel, which in this case is a fathah.  The six vowel count medd is then allowed for the first way, recognizing the original sukoon on the , and two vowel count medd allowed in the second way allowing for  the  fact that the is no longer saakinah so the medd is dropped.  

Wa iyyaakum.
Wa assalaam alaikum wa rahmatullah