If you're talking about strong convergence, your statement that the ASS converges to a quotient of $\pi_*(Y)$ is not correct, even for $Y=S^0$ and $E=H\mathbb{F}_p$, where it converges to the p-adic homotopy groups $\pi_*(S^0) \otimes \mathbb{Z}_p$. Which agrees with the localization in this case. Weak convergence you do have, but that's really less than useful.

1. For spectra, this is a correct characterization of localizations; for spaces, not quite. A space Y is E-local if for every E-equivalence $X \to X'$, the induced map $[X',Y] \to [X,Y]$ is a bijection. An $E$-equivalence, of course, is a map that induces an isomorphism in $E$-homology.

2. I don't have a copy of Switzer handy, but the kind of homology theory $E$ in the proposition (connective with $\pi_0 = \mathbb{Z}$) will produce exactly the same kind of localizations as singular homology with coefficients in $\mathbb{Z}$ if your spectrum $Y$ is connective, as you assume; this is proved, I think, in Bousfield's original papers on localization. In other words, the localization is the identity.

It is not in general true that the Adams spectral sequence converges in any sense to the localization, though. It converges to the completion (nilpotent completion, to be exact) in the sense of Ravenel 1984. And if you're into convergence criteria, the kind of convergence is conditional convergence, in particular if there's no derived $E^\infty$ term then it converges strongly. Unfortunately, the nilpotent completion is only an approximation to the localization. The nilpotent completion of a connective spectrum at $H\mathbb{Z}$, however, is the spectrum itself; this is a Postnikov tower argument. So no problem in that case, and no contradiction to Switzer's proposition.

3. $[L_E X,L_E Y] = [X,L_E Y]$ by the universal property. For finite spectra $X$, the ASS behaves pretty much the same as for $X=S^0$. For infinity $X$, I believe the situation becomes more complicated, but I'm not sure about the details.

If you're talking about strong convergence, your statement that the ASS converges to a quotient of $\pi_*(Y)$ is not correct, even for $Y=S^0$ and $E=H\mathbb{F}_p$, where it converges to the p-adic homotopy groups $\pi_*(S^0) \otimes \mathbb{Z}_p$. Which agrees with the localization in this case. Weak convergence you do have, but that's really less than useful.

1. For spectra, this is a correct characterization of localizations; for spaces, not quite. A space Y is E-local if for every E-equivalence $X \to X'$, the induced map $[X',Y] \to [X,Y]$ is a bijection. An $E$-equivalence, of course, is a map that induces an isomorphism in $E$-homology.

2. I don't have a copy of Switzer handy, but the kind of homology theory $E$ in the proposition (connective with $\pi_0 = \mathbb{Z}$) will produce exactly the same kind of localizations as singular homology with coefficients in $\mathbb{Z}$ if your spectrum $Y$ is connective, as you assume; this is proved, I think, in Bousfield's original papers on localization. In other words, the localization is the identity.

It is not in general true that the Adams spectral sequence converges in any sense to the localization, though. It converges to the completion (nilpotent completion, to be exact) in the sense of Ravenel 1984. And if you're into convergence criteria, the kind of convergence is conditional convergence, in particular if there's no derived $E^\infty$ term then it converges strongly. Unfortunately, the nilpotent completion is only an approximation to the localization. The nilpotent completion of a connective spectrum at $H\mathbb{Z}$, however, is the spectrum itself; this is a Postnikov tower argument. So no problem in that case, and no contradiction to Switzer's proposition.

3. $[L_E X,L_E Y] = [X,L_E Y]$ by the universal property. For finite spectra $X$, the ASS behaves pretty much the same as for $X=S^0$. For infinite $X$, I believe the situation becomes more complicated, but I'm not sure about the details.