You have to restate the meaning of \mathbb:
\AtBeginDocument{
\DeclareSymbolFont{AMSb}{U}{msb}{m}{n}
\DeclareSymbolFontAlphabet{\mathbb}{AMSb}}
This should go somewhere after loading mathpazo.
The pplj family has no slanted font: indeed the relevant line in t1pplj.fd reads
\DeclareFontShape{T1}{pplj}{m}{sl}{<->ssub * pplj/m/it}{}
If you want to avoid the messages, you can write
\DeclareFontShape{T1}{qpl}{m}{sl} { <-> ssub * qpl/m/it }{}
\DeclareFontShape{T1}{qpl}{b}{sl} { <-> ssub * qpl/b/it }{}
so getting Pagella Italic. The most convenient way is just not using slanted, because it doesn't exist.
If I remove the scale=0.95 option from the call to tgpagella, I get characters that are the same width as the normal Palatino provided by mathpazo. If you want smaller fonts, just use 9pt as the main size.
I see bad spacing in the second math formula only because you say \left(...\right), which in this case is wrong.
There is no need to do \RequirePackage before \documentclass:
\documentclass[a4paper, oneside, 9pt, extrafontsizes, showtrims, draft]{memoir}
\usepackage[T1]{fontenc} % choose the default encoding
\usepackage[english]{babel} % choose the language
\usepackage[sc]{mathpazo} % use mathpazo for math fonts
\usepackage{tgpagella} % but use tgpagella as main font
\usepackage[scaled=0.75]{luximono}
% Are the following really necessary? I don't think so
%\renewcommand*{\memfontfamily}{qpl} % tgpagella as main memoir font
%\renewcommand*{\memfontenc}{T1}
%\renewcommand*{\memfontpack}{tgpagella}
\normalfont % we want to avoid annoying warnings
\DeclareFontShape{T1}{qpl}{m}{sl} { <-> ssub * qpl/m/it }{}
\DeclareFontShape{T1}{qpl}{b}{sl} { <-> ssub * qpl/b/it }{}
\linespread{1.05} % add something to the interline skip
\setlrmarginsandblock {30mm}{*}{*}
\setlxvchars \setxlvchars[\small\sffamily]
\checkandfixthelayout
\fixpdflayout
\usepackage[final, babel=true]{microtype}
\usepackage{amsmath} % amsmath which also loads fonts?
\nouppercaseheads
\pagestyle{headings}
Best Answer
A sans serif font that blends well with Palatino (and is freely available) is Bera Sans:
URLs are often typeset in a monospaced font. Probably anything is better than Courier; I recommend Inconsolata:
(The specified scale factors will adapt the x-height of the respective font to that of the Palatino.)
EDIT: The Myriad font and its Open Type incarnation Myriad Pro, which are not freely available, are even better sans serif matches for Palatino. The files (excluding the actual font files) necessary to use Myriad with LaTeX can be found here.