Problem 1:
Each name consists of four parts: First, von, Last, and Jr; each part consists of a (possibly empty) list of name-tokens. As usually there are only two parts (first name/s and surname/s) automatic identification of these parts fail when parts are not (1) in the correct order, or (2) when there are more than one space, and then is unclear for BibTeX which one is a true limit between parts. This is solved enclosing the components of the same part in braces and/or using commas to allow BibTeX's identification of named components. So you can write "First Last" or "Last, First", "First von Last" or "von Last, First" and with Jr. "von Last, Jr, First" or "Last, Jr, First"
In your example, you can enter "Hall, Jr., Marshal", but if you want include "Jr." after the fist name then you can use "Hall, {Marsall Jr.}"
Problem 2:
In am not sure if you simply want to render $elle$ in the PDF as is. In this case write \$elle\$ in Jabref (as well as inside ERT boxs in LyX text).
Problem 3:
This is not a Jabref problem, is a "feature" of your bibliographic style. The "problem" could be in articles, for example, but not in books. To avoid this you can change to another style that have not this feature, or modify the .bst style file, or make your own style for example with the custom-bib package, or switch to biblatex, or simply protect with braces {} the capitals of proper names, technical acronyms, initials of subtitles, etc.
This last solution is self-explained in this plain LaTeX file, as well as solutions to problems 1 and 2:

\documentclass{article}
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@article{marshallone,
author = "Hall, {Marshall Jr.}",
title = "About ACRONYMS, {A}CRONYMS and {ACRONYMS}",
journal=''Some Journal'',
volume=''3'',
pages=''1--2'',
year = 2013
}
@BOOK{Marchall1979,
author = "Hall, Jr., Marshall and E. B. Sheriff",
title = "The Elements of Style of \$elle\$'",
edition = "Third",
year = 1979
}
\end{filecontents}
\begin{document}
\nocite{*}
\bibliographystyle{plain}
\bibliography{\jobname}
\end {document}