I answer in order to all your questions
Every field in biblatex can be modified by the command DeclareFieldFormat. To change settins only at cite commands you can use the command \AtEveryCite.
\AtEveryCite{%
\DeclareFieldFormat{url}{}%
\DeclareFieldFormat{urldate}{}%
\DeclareFieldFormat{doi}{}%
}
Instead you can also use AtEveryBibitem.
The word "in" can be removed by redefining the macro in: as follow:
\renewbibmacro*{in:}{}
Extra line breaks can be done by expanding the relevant field formats.
\DeclareFieldFormat{url}{\newline\mkbibacro{URL}\addcolon\space\url{#1}}
\DeclareFieldFormat{doi}{\newline%
\mkbibacro{DOI}\addcolon\space
\ifhyperref
{\href{http://dx.doi.org/#1}{\nolinkurl{#1}}}
{\nolinkurl{#1}}}
Every definition can be found in the file biblatex.def
Here the complete code:
\documentclass[]{article}
\usepackage{filecontents}
\begin{filecontents}{References.bib}
@article{Sarukkai:2005,
author = {Sarukkai, S},
title = {Revisiting the 'unreasonable effectiveness' of mathematics},
journal = {Current science},
year = {2005},
url = {http://www.ias.ac.in/currsci/feb102005/415.pdf},
}
\end{filecontents}
\usepackage[style=verbose]{biblatex}
\addbibresource{References.bib}
\AtEveryCite{%
\DeclareFieldFormat{url}{}%
\DeclareFieldFormat{urldate}{}%
\DeclareFieldFormat{doi}{}%
}
\DeclareFieldFormat{url}{\newline\mkbibacro{URL}\addcolon\space\url{#1}}
\DeclareFieldFormat{doi}{\newline%
\mkbibacro{DOI}\addcolon\space
\ifhyperref
{\href{http://dx.doi.org/#1}{\nolinkurl{#1}}}
{\nolinkurl{#1}}}
\renewbibmacro*{in:}{}
\begin{document}
This, \fullcite{Sarukkai:2005} includes the URL for the citation, which I don't want; thought I do want it in the bibliography.
\printbibliography
\end{document}
