The superfluous space comes from an uncommented line end in the bibmacro maintitle+volumes+parts+. The double punctuation comes from a bare \bibsentence in cite:opcit. The first is easily fixed by adding a % (What is the use of percent signs (%) at the end of lines? (Why is my macro creating extra space?)), for the second issue I'm not sure why the \bibsentence is there in the first place, so the output with my workaround probably needs some careful scrutiny.
\documentclass{article}
\usepackage[backend=biber,style=gost-inline]{biblatex}
\renewbibmacro{maintitle+volumes+parts+}[1]{%
\iffieldsequal{maintitle}{#1}
{\clearfield{maintitle}%
\clearfield{mainsubtitle}%
\clearfield{maintitleaddon}}
{\iffieldundef{maintitle}
{\usebibmacro{#1}%
\setunit{\addcolondelim}%
\usebibmacro{involumes+otherparts}{\setunit{\addcomma\space}}%
\newunit
\usebibmacro{volume+parts}{\newunit}}
{\usebibmacro{maintitle}%
\newunit
\usebibmacro{involumes+otherparts}{\setunit*{\addcomma\space}}%
\newunit
\usebibmacro{volume+parts}{\newunit}%
\newunit
\usebibmacro{#1}}}%
\newunit}
\renewbibmacro{cite:opcit}{%
\ifnameundef{labelname}
{\printfield{label}}
{\printnames{labelname}}%
\setunit{\labelnamepunct}%
\printtext[bibhyperlink]{%
\bibstring[\mkibid]{opcit}}}
\begin{filecontents}{\jobname.bib}
@book{Millimetron,
author = {John Millimetron},
title = {A title of the first book},
}
@book{Ilusive,
author = {Joe Elusive},
location = {Lipsiae},
title = {A title of the second book},
year = {1871},
}
\end{filecontents}
\addbibresource{\jobname.bib}
\begin{document}
Here \autocite{Millimetron} is some \autocite{Ilusive} text \autocite{Millimetron}.
\printbibliography
\end{document}

I reported the issues at https://github.com/odomanov/biblatex-gost/issues/29 and https://github.com/odomanov/biblatex-gost/issues/30.
Update
The biblatex-gost maintainer has quickly resolved the two issues above and released version 1.20 of the style bundle.
The problem with your new entry is another missing %, this time in the driver for @unpublished. It has already been added (https://github.com/odomanov/biblatex-gost/commit/8d829bfcb854655f4ee7a3eb622bd422d50083a8), but in the meantime you can add the following to your preamble.
\documentclass{article}
\usepackage[backend=biber,style=gost-inline]{biblatex}
\renewbibmacro{maintitle+volumes+parts+}[1]{%
\iffieldsequal{maintitle}{#1}
{\clearfield{maintitle}%
\clearfield{mainsubtitle}%
\clearfield{maintitleaddon}}
{\iffieldundef{maintitle}
{\usebibmacro{#1}%
\setunit{\addcolondelim}%
\usebibmacro{involumes+otherparts}{\setunit{\addcomma\space}}%
\newunit
\usebibmacro{volume+parts}{\newunit}}
{\usebibmacro{maintitle}%
\newunit
\usebibmacro{involumes+otherparts}{\setunit*{\addcomma\space}}%
\newunit
\usebibmacro{volume+parts}{\newunit}%
\newunit
\usebibmacro{#1}}}%
\newunit}
\renewbibmacro{cite:opcit}{%
\ifnameundef{labelname}
{\printfield{label}}
{\printnames{labelname}}%
\setunit{\labelnamepunct}%
\printtext[bibhyperlink]{%
\bibstring[\mkibid]{opcit}}}
\DeclareBibliographyDriver{unpublished}{%
\usebibmacro{bibindex}%
\usebibmacro{begentry}%
\usebibmacro{heading}%
\newunit
\usebibmacro{author}%
\setunit{\labelnamepunct}%
\usebibmacro{title}%
\setunit{\addcolondelim}%
\usebibmacro{translation}%
\def\bbx@gost@respdelim{\setunit{\respdelim}}% ----- Resp starts -----
\usebibmacro{byauthor}%
\setunit*{\resppunct}%
\usebibmacro{credits}%
\newunit\newblock
\printlist[semicolondelim]{specdata}%
\newunit\newblock
\usebibmacro{location+date}%
\newunit\newblock
\usebibmacro{isbn}%
\newunit\newblock
\usebibmacro{url+urldate+note}%
\newunit
\printfield{howpublished}%
\newunit\newblock
\usebibmacro{addendum+pubstate}%
\setunit{\bibpagerefpunct}\newblock
\usebibmacro{pageref}%
\newunit\newblock
\usebibmacro{relatedblock}%
\usebibmacro{finentry}}
\begin{filecontents}{\jobname.bib}
@book{Millimetron,
author = {John Millimetron},
title = {A title of the first book},
}
@book{Ilusive,
author = {Joe Elusive},
location = {Lipsiae},
title = {A title of the second book},
year = {1871},
}
@Unpublished{Antonin,
author = {Antonin Kapustin},
title = {Katalog of manuscripts of the Sinai monastery},
year = {1870},
location = {Sinai},
}
\end{filecontents}
\addbibresource{\jobname.bib}
\begin{document}
Here \autocite{Millimetron} is some \autocite{Antonin} text \autocite{Millimetron}.
\printbibliography
\end{document}