I answer in order to all your questions
Fields in biblatex can be deleted by the command \clearfield/\clearlist (you can consult the biblatex documentation to find out whether a field is a field or a list, refer to §2.2.2 Data Fields, pp. 15-24). To influence only cite commands you can use the command \AtEveryCitekey.
\AtEveryCitekey{%
\clearfield{url}%
\clearfield{urlyear}
\clearfield{doi}%
}
Instead you can also use AtEveryBibitem for the bibliography.
The word "in" can be removed by redefining the macro in: as follows:
\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}
\AtEveryCitekey{%
\clearfield{url}%
\clearfield{urlyear}
\clearfield{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}
