Switching from amsrefs to biblatex already shows how you can make the mrnumber show in the bibliography, so here we show mrclass and mrreviewer as well.
The idea is the same. First we define a new datamodel file (.dbx) with the new fields and load that with the datamodel option. Then we need formats for those fields and finally we need to print the fields in a bibmacro (mrinfo) that is called in one of the standard macros (doi+eprint+url).
\documentclass[british]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{babel}
\usepackage{csquotes}
\usepackage{filecontents}
\begin{filecontents*}{mrnumber.dbx}
\DeclareDatamodelFields[type=field,datatype=literal]{mrnumber,mrclass}
\DeclareDatamodelFields[type=list, datatype=name]{mrreviewer}
\DeclareDatamodelEntryfields{mrnumber,mrclass,mrreviewer}
\end{filecontents*}
\usepackage[style=numeric, backend=biber, sorting=nyt, datamodel=mrnumber]{biblatex}
\usepackage{hyperref}
\DeclareFieldFormat{mrnumber}{%
\ifhyperref
{\href{http://www.ams.org/mathscinet-getitem?mr=1#1}{MR#1}}
{MR#1}}
\DeclareFieldFormat{mrclass}{#1}
\DeclareNameAlias{bymrreviewer}{byeditor}
\newbibmacro*{mrinfo}{%
\printfield{mrnumber}%
\iffieldundef{mrclass}
{\setunit*{\addcomma\space}}
{\setunit*{\addspace}}%
\printfield{mrclass}%
\setunit*{\addcomma\space}%
\ifnameundef{mrreviewer}
{}
{\bibstring{byreviewer}%
\setunit{\addspace}%
\printnames[bymrreviewer]{mrreviewer}}}
\newtoggle{bbx:mrinfo}
\DeclareBibliographyOption[boolean]{mrinfo}[true]{\settoggle{bbx:mrinfo}{#1}}
\ExecuteBibliographyOptions{mrinfo}
\renewbibmacro*{doi+eprint+url}{%
\iftoggle{bbx:doi}
{\printfield{doi}}
{}%
\newunit\newblock
\iftoggle{bbx:mrinfo}
{\usebibmacro{mrinfo}}
{}%
\newunit\newblock
\iftoggle{bbx:eprint}
{\usebibmacro{eprint}}
{}%
\newunit\newblock
\iftoggle{bbx:url}
{\usebibmacro{url+urldate}}
{}}
\begin{filecontents*}{\jobname.bib}
@article{gratzer,
author = {Grätzer, G.},
title = {The order of principal congruences of a bounded lattice},
journal = {Algebra Universalis},
volume = {70},
year = {2013},
number = {1},
pages = {95--105},
mrclass = {06B10 (06A06)},
mrnumber = {3073941},
mrreviewer = {S. Parameshwara Bhatta},
doi = {10.1007/s00012-013-0242-3},
}
\end{filecontents*}
\addbibresource{\jobname.bib}
\addbibresource{biblatex-examples.bib}
\begin{document}
\cite{gratzer,sigfridsson}
\printbibliography
\end{document}

For the second issue I suggest you ask a new question since that is not related to biblatex at all. You could query https://mathscinet.ams.org/mrlookup with your .bib entries and get the result. As long as your author and title fields are accurate (and coincide with the info on MathSciNet) that should give you the correct entry. The Python tool https://pypi.org/project/bibupdate/ seems to do just that.
biblatexdoes not knowmrnumberand friends, but you can tell it to recognise those fields, see https://tex.stackexchange.com/q/306908/35864. – moewe Mar 04 '19 at 21:33biblatexquestion and I encourage you to ask it in a separate question. I suspect this might be possible if you somehow query MathSciNet with your old.bibfiles and retrieve the relevant information. But there is a risk of false matches and incorrect information there, so I would go for a manual approach when needed so I can double check stuff.{ ... }around your\usepackage{biblatex}and\addbibresourcestatements in your document, but let me just warn future visitors that that would most likely not work. Loading packages in a group is quite likely to go wrong, because packages normally assume that all definitions they make are global in scope. – moewe Mar 05 '19 at 07:10