2 added 177 characters in body

EDIT 20-03-12 It seems from the recent answers of Douglas Somerset and Ulrich Pennig that what I claim below is false, and so this answer should be "dis-accepted".

I think (although I admit I don't know the details) that the answer to both questions is yes, by a theorem of Dauns and Hoffman. According to the version quoted in the article

T. Becker, A few remarks on the Dauns-Hofmann theorems for $C^\ast$-algebras. Archiv der Mathematik 43 (1984) no. 3, 265-269 [Math Review]

$A$ can be realized as the algebra of continuous sections of some kind of continuous $C^\ast$-algebra-bundle with base space $T$.

However, since I am not a specialist, I may have misread or misunderstood.

1

I think (although I admit I don't know the details) that the answer to both questions is yes, by a theorem of Dauns and Hoffman. According to the version quoted in the article

T. Becker, A few remarks on the Dauns-Hofmann theorems for $C^\ast$-algebras. Archiv der Mathematik 43 (1984) no. 3, 265-269 [Math Review]

$A$ can be realized as the algebra of continuous sections of some kind of continuous $C^\ast$-algebra-bundle with base space $T$.

However, since I am not a specialist, I may have misread or misunderstood.