Just needed to know the correct way of reciting as in joinin the muhammadin wa aale muhammad is wrong?

And what is confusing is in arabic its written as Muhammadinw wa aale Muhammad. Does the tanwin rule apply here in the recitation of salwat or is it recitied as you mentioned above Muhammad wa aale Muhammad

Joining is correct and you can say it without tanween also is correct. Both ways are acceptable and correct.