’Assalaamu ‘alaikum wa rahmatul-laahi wa barakaatuh,

When the taa’ or kaaf are mushaddad, is the first letter of the shaddah still pronounced with the characteristic of hams, or how? Jazaakal-laahu khairaa.

Was-salaamu ‘alaikum wa rahmatul-laahi wa barakaatuh.

Wa Alaikum Assalam Wa Rahmatullahi Wa Barakatuh,
  The characteristic of and are always present in the two letters even when the letter has a vowel.  These characteristics are intrinsic and never leave the letter.  When  when these two letters have a sukoon however, then the characteristics of shidda and hams are applied sequentially, meaning one after the other, first and second  .
In the case of the shaddah, or empahisized letter, even though the first kaaf and ta' of the shaddah have a sukoon, both the characteristics will be applied at the same time (simulataneously) and not sequentially.  That is because even though the letter has the mechanism of collision for a sukoon and separation for a vowel, we do not apply rules on the saakin letter of a shaddah. 
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Wa iyyaaakum wa-l-muslimeen. 
Wa asslaam alaikum wa rahmatullahi wa barakatuh.