Does Household final consumption Expenditure include Consumption expenditure of Imported Goods
Yes.

Household final consumption expenditure consists of the expenditure, including expenditure whose value must be estimated indirectly, incurred by resident households on individual consumption goods and services, including those sold at prices that are not economically significant and including consumption goods and services acquired abroad.
(From System of National Accounts, 2008, p. 623, PDF).

To elaborate, in the familiar identity $$Y = C + I + G + (X - M),$$ the reason we subtract $M$ (imports) is that the terms $C$, $I$, and $G$ also include imported goods and services.
In particular and to your secondary question, let:

$C_d =$ Consumption of domestically-produced goods and services.
$C_m =$ Consumption of foreign-produced goods and services.

Then: $$C=C_d+C_m.$$
And what I believe you want is $\frac{C_d}{C}$ or equivalently $1-\frac{Cm}{C}$.