Cat initially resents everyone embracing new dog (7)
I believe the answer is:
ragdoll
'cat' is the definition.
(I know that Ragdoll is a type of cat)
'initially resents everyone embracing new dog' is the wordplay.
'initially' suggests taking the first letters.
'everyone' becomes 'all' (all people).
'embracing' means one lot of letters goes inside another.
'new' is an anagram indicator.
The initial letter of 'resents' is 'r'.
'dog' anagrammed gives 'gdo'.
'all' placed around 'gdo' is 'agdoll'.
'r'+'agdoll'='RAGDOLL'
(Other definitions for ragdoll that I've seen before include "cat" , "Toy made with scraps of cloth etc" .)