Yes, it is possible to link to MathSciNet. This can be done as in Biblatex and Pubmed/Pubmed Central IDs. Either as a manifestation of the eprint feature via
\DeclareFieldFormat{eprint:mrnumber}{%
MR\addcolon\space
\ifhyperref
{\href{http://www.ams.org/mathscinet-getitem?mr=#1}{\nolinkurl{#1}}}
{\nolinkurl{#1}}}
and then
eprint = {1678525},
eprinttype = {mrnumber},
in the .bib file.
Or as a separate field with a new datamodel. Call it mrnumber.dbx
\DeclareDatamodelFields[type=field,datatype=verbatim]{mrnumber}
\DeclareDatamodelEntryfields{mrnumber}
then load that datamodel with the option datamodel=mrnumber, specify the formatting
\DeclareFieldFormat{mrnumber}{%
MR\addcolon\space
\ifhyperref
{\href{http://www.ams.org/mathscinet-getitem?mr=#1}{\nolinkurl{#1}}}
{\nolinkurl{#1}}}
and print it
\renewbibmacro*{doi+eprint+url}{%
\iftoggle{bbx:doi}
{\printfield{doi}}
{}%
\newunit\newblock
\printfield{mrnumber}%
\newunit\newblock
\iftoggle{bbx:eprint}
{\usebibmacro{eprint}}
{}%
\newunit\newblock
\iftoggle{bbx:url}
{\usebibmacro{url+urldate}}
{}}
You can see this in action in the example at the end.
It is very easily possible to make book titles bold
\DeclareFieldFormat[book]{title}{\mkbibbold{#1}}
I don't know of an option to make the output look like amsrefs with only a few strokes on your keyboard. But I believe that many changes needed to the standard biblatex styles can be facilitated easily. Of course there might be the odd detail that is harder to implement.
If you need features that are only present in biblatex or significantly harder to get to work in amsrefs, a switch seems reasonable. But if you don't miss anything or can work around the deficiencies reasonably easy, you don't need to go through the ordeals of switching.
See also When should I use amsrefs instead of regular bibtex? and How popular is amsrefs in comparison with Bibtex?
Example
\documentclass[british]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{babel}
\usepackage{csquotes}
\usepackage{filecontents}
\begin{filecontents*}{mrnumber.dbx}
\DeclareDatamodelFields[type=field,datatype=verbatim]{mrnumber}
\DeclareDatamodelEntryfields{mrnumber}
\end{filecontents*}
\usepackage[backend=biber, style=numeric, datamodel=mrnumber]{biblatex}
\usepackage{hyperref}
\begin{filecontents*}{\jobname.bib}
@article{grabiner,
author = {David J. Grabiner},
title = {Brownian motion in a Weyl chamber, non-colliding particles, and random matrices},
journal = {Annales de l'Institut Henri Poincare (B) Probability and Statistics},
volume = {35},
number = {2},
pages = {177-204},
year = {1999},
doi = {10.1016/S0246-0203(99)80010-7},
mrnumber = {1678525},
}
\end{filecontents*}
\DeclareFieldFormat{mrnumber}{%
MR\addcolon\space
\ifhyperref
{\href{http://www.ams.org/mathscinet-getitem?mr=#1}{\nolinkurl{#1}}}
{\nolinkurl{#1}}}
\renewbibmacro*{doi+eprint+url}{%
\iftoggle{bbx:doi}
{\printfield{doi}}
{}%
\newunit\newblock
\printfield{mrnumber}%
\newunit\newblock
\iftoggle{bbx:eprint}
{\usebibmacro{eprint}}
{}%
\newunit\newblock
\iftoggle{bbx:url}
{\usebibmacro{url+urldate}}
{}}
\DeclareFieldFormat[book]{title}{\mkbibbold{#1}}
\addbibresource{\jobname.bib}
\addbibresource{biblatex-examples.bib}
\begin{document}
\cite{cicero,grabiner,sigfridsson}
\printbibliography
\end{document}

amsrefshad a big potential, in my opinion, but it's not developed nor maintained;biblatexhas several styles and is actively developed. – egreg Apr 28 '16 at 20:16amsstyleand get the defaults?' But if you mean 'can I configure it to use those defaults?' then probably. – cfr Apr 28 '16 at 20:46