I think I have a positive answer for (1). Let $p > 3$ be a prime, let $\\mathbbf{K}_1 = \mathbbf{Q}[ \zeta_p ]$$\mathbb{K}_1 = \mathbb{Q}[ \zeta_p ]$ be the $p$-cycotomic extension and let $q = 1 \bmod p$ be some rational prime. Let $\\mathbbf{F} \subset \\mathbbf{Q}[ \zeta_q ]$$\mathbb{F} \subset \mathbb{Q}[ \zeta_q ]$ be the subfield of degree $p$ over $\\mathbbf{Q}$$\mathbb{Q}$. Obviously, $\K_1$$\mathbb{K}_1$ has $r_2+1 = (p+1)/2$ independent $\Z_p$$\mathbb{Z}_p$-extensions. Let $\\mathbbf{M}$$\mathbb{M}$ be the composite of all $\\mathbbf{Z}_p$$\mathbb{Z}_p$-extensions of $\\mathbbf{K}_1$$\mathbb{K}_1$ and $D(q) \subset X = Gal(\mathbbf{M}/\\mathbbf{K}_1)$$D(q) \subset X = Gal(\mathbb{M}/\mathbb{K}_1)$ be the decomposition group of some prime of $\\mathbbf{K}_1$$\mathbb{K}_1$ above $q$ - it is isomorphic to $\mathbbf{Z}_p$$\mathbb{Z}_p$. So we can choose some $\mathbbf{Z}_p$$\mathbb{Z}_p$ - extension of $\mathbbf{K}_1$$\mathbb{K}_1$ which is fixed by $D(q)$; let this be $\mathbbf{L}$$\mathbb{L}$. Then $q$ is totally split in $\mathbbf{L}/\mathbbf{Q}$$\mathbb{L}/\mathbb{Q}$. Let now \mathbbf{K}' = \mathbbf{K}_1$\mathbb{K}' = \mathbb{K}_1 . \mathbb{F}$ and let $\mathbb{L}' = \mathbb{L} . \mathbb{F}$. \mathbbf{F}$ and let $\mathbbf{L}' = \mathbbf{L}It is an exercise in the application of the ambig ideal lemma of Chevalley, to adapt Iwasawa's seminal proof to the present situation and show that $\mu(\mathbb{L}') > 0$. \mathbbf{F}$. It is an exercise in the application of the ambig ideal lemma of Chevalley, to adapt Iwasawa's seminal proof to the present situation and show that $\mu(\mathbbf{L}') > 0$. Of course, $\mathbbf{K}'$Of course, $\mathbb{K}'$ is a (quite simple) cyclotomic field, which answers your question in the affirmative.
For extensions $\mathbbf{K}'/\mathbbf{K}$$\mathbb{K}'/\mathbb{K}$ of degree not divisible by $p$, the method of Iwasawa fails. So the question (2) appears to be difficult ...
(I only now saw the question of Filippo. The answers were in the text, but maybe TeX helps...)
