Point (1) is correctly addressed in the comment by Hoot. As for point (2), your intuition is on the right track. On the other hand, you should keep track of the multiplicities of the loci involved.
As a matter of example, let $X$ be the projective plane, $Y$ the cuspidal rational curve, and $Z$ the singular point of $Y$. $Z$ is a regular subvariety of $X$, so the exceptional divisor is just a copy of $\mathbb{P}^1$ (in general, if you blow up something singular, the exceptional locus might be pretty ugly though). The strict transform of $Y$ (i.e. $Y'$ in your notation) is going to be a smooth rational curve tangent to $E$. This reflects that $Y$ has multiplicity 2 along $Z$. This gives you that $\pi^*Y= Y'+2E$. As you see, the ingredients are exactly the ones you expected, but, in this case, they are weighted with coefficients depending on the singularities of $Y$ along $Z$.
Edit I am reading your answer more carefully now. If both $Y$ and $Z$ are smooth, then claim (2) is fine as well.
Addendum You comment is right. The blow up is an isomorphism over $X \setminus Z$. In particular, if $\widehat{Y}$ is disjoint from $Z$, its strict transform $\widehat{Y}'$ coincides with the pullback $\pi^*(\widehat{Y})$ , and it is isomorphic to $\widehat{Y}$. Now, if $Y$ and $\widehat{Y}$ are linearly equivalent, so are their pullbacks (just because the isomorphism between $\mathcal{O}_X(Y)$ and $\mathcal{O}_X(\widehat{Y})$ induces an isomorphism between their pullbacks). On the other hand, this is telling you that the strict transforms of linear equivalent divisors are not linearly equivalent if just one of the two goes through $Z$.
Let me be more explicit. Blow up a point $P$ in $\mathbb{P}^2$. Let $L_1$ be a line through $P$, and $L_2$ a line not containing $P$. Denote by $M_1$ and $M_2$ the respective strict transforms. Then, by what above said, we have $\pi^*L_1=M_1+E$, and $\pi^*L_2=M_2$. By Bezout theorem we know that the intersection products $L_1 \cdot L_2=(L_1)^2=(L_2)^2=1$. In particular $L_1$ and $L_2$ meet properly at one point, say $Q$. Now, since $L_2$ does not go through $P$, we have that the pullbacks $M_1+E$ and $M_2$ meet properly at one point (the only premiage of $Q$). Given that these divisors are also linearly equivalent to each other, we get $1=(M_2)^2=M_2 \cdot (M_1+E)=(M_1+E)^2$. In particular, we get $1=(M_1+E)^2=M_1^2+2M_1\cdot E+ E^2$. Since $M_1$ and $E$ meet properly at one point, we know $M_1 \cdot E=1$. Then, we know that $E^2=\mathrm{deg}\mathcal{O}_{X'}(E)_{|E}$. By the description in sections 7 and 8 in chapter 2 of Hartshorne, we know that this is the relative $\mathcal{O}(1)$ bundle, that $E=\mathbb{P}^1$; these things together tell us that $\mathcal{O}_{X'}(E)_{|E}\cong \mathcal{O}_{\mathbb{P}^1}(-1)$. Thus, that degree is $-1$, so $E^2=-1$. This is negative self intersection is phrased as $E$ does not deform: there is no other effective divisor equivalent to $E$. Now, putting this in our previous equation, we get that $(M_1)^2=0$. As you see, $(M_1)^2 \neq (M_2)^2$; in particular, they can not be linearly equivalent.
Best Answer
This is a community wiki post consisting of the answer from the comments, in order to remove this question from the unanswered queue.
Here is the statement of the proposition (with some clarifying language on the assumptions):