Non-movable feasts in the Lent and Easter seasons The latest possible date for Ash Wednesday is March 10th and the earliest possible date for Pentecost is May 10th.  Besides the movable feasts, is there any indication that the Catholic Church places those feasts inside of these seasons for specific reasons?  

For instance, the Solemnity of St. Joseph is March 19th, this date is always in Lent (although the feast may be transferred to Easter), yet it is a Holy Day of Obligation in many places, (generally means it is equivalent with a Sunday in saying "you need to go to Church today").  Is it, and other feasts, placed inside Lent or Easter to further the purposes that their respective seasons seek to celebrate?
It has to do with, as with the Orthodox, the position of Easter (Pascha) which is a reflection of the calculation of the Passover prior to Pentecost. 

Given the common tradition in both Rome and the East on this, at least for Ash Wednesday and Pentecost, it has all to do with their positions vis a vis Easter. It says the 'last possible' or 'first possible' because - lets say Ash Wednesday is the Wednesday in the week seven weeks before Easter - since Easter cannot be earlier than a certain date, nor can Ash Wednesday. The same logic follows for Pentecost, which is - going back before Christ - the fifty-day after Passover celebration. 

As for St. Joseph's solemnity I can't say, I don't know the history (this feast is not in Orthodoxy.) Generally the explanation for any fixed feast (which can either be fixed against the date of Easter or the date of the First of the Year) is entirely historical.

Inclusion of feasts specifically in Lent, as for instance in our tradition, The Sunday of Orthodoxy, or the Sunday of St. Mary of Egypt, has much to do with the catechetical, ascetical character of Lent itself and special lenten tide feasts reflect that.

Annunciation on the other hand, sometimes in the Old Calendar can occur ON EASTER. This is called Kyriepascha, and you can't move Annunciation or Pascha. So you DO ALL THE PRAYERS. Which means all the texts specific to each feast are done, and considering how strange Pascha is structured, it's a crazy service!