You haven't given us an awful lot to go by, but the first issue looks like it could be Set limit to one author when using "et al." in biblatex. biblatex tries to make even truncated lists of name unique, so that Rakel Hönnudóttir and Harpa Þorsteinsdóttir and Elísa Viðarsdóttir and Rakel Hönnudóttir and Harpa Þorsteinsdóttir and Anna Björk Kristjánsdóttir are not both abbreviated to Rakel Hönnudóttir et al. That feature can be turned off with uniquelist=false.
A second unique feature, uniquename, may cause the first name to appear in some citations and not in others, see biblatex, authoryear style: In-text citations display first name initials for certain bibliography entries, Biblatex: first names in text only for some sources - why? and Literature with Biber generates strange citations: firstnames appear erratically.
For the other issue let me explain how biblatex (and BibTeX) treat names.
For BibTeX/Biber a name can consist of four parts, the most important parts being first name and last name. Almost all styles I know operate under the assumption that the first name is a given name and the last name is a family name. In newer versions of biblatex this is cemented by officially designating the name parts as family and given name. This view and the entire name scheme is largely influenced by Anglo-American conventions and works well for many Western names and also fairly well for East Slavic conventions (at least the Russian form).
Furthermore in the Anglo-American world, the family name is primarily used to identify an individual. The given name or an initial can be added to avoid ambiguity, but this is not always done.
So if you enter an Icelandic name such as
author = {Katrín Jakobsdóttir},
Katrín is correctly identified as the given name, but Jakobsdóttir is treated as a family name. In particular biblatex will primarily use Jakobsdóttir to refer to that person.
An easy solution is to wrap the name in braces when giving it in the .bib file.
author = {{Katrín Jakobsdóttir}},
and (only wrap individual names in braces, not several names at once!)
author = {{Katrín Jakobsdóttir} and {Jóhanna Sigurðardóttir}},
that way Katrín Jakobsdóttir and Jóhanna Sigurðardóttir are parsed as the family name resulting in use of the full name in citations. The name will be treated as one unit and will not be displayed in reversed order as *Jakobsdóttir, Katrín.
\documentclass[icelandic]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{babel}
\usepackage{csquotes}
\usepackage[style=authoryear, backend=biber, mincitenames=1, maxcitenames=1, maxbibnames=999]{biblatex}
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@book{four,
author = {{Katrín Jakobsdóttir}},
title = {Four},
date = {1984},
}
@book{five,
author = {{Jóhanna Sigurðardóttir}},
title = {Five},
date = {1985},
}
\end{filecontents}
\addbibresource{\jobname.bib}
\begin{document}
\cite{four}
\cite{five}
\printbibliography
\end{document}
If you only have Icelandic names in your bibliography or want to treat foreign names in the Icelandic fashion as well, you could instead redefine the name format and handling a bit.
\documentclass[icelandic]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{babel}
\usepackage{csquotes}
\usepackage[style=authoryear, backend=biber, mincitenames=1, maxcitenames=1, maxbibnames=999]{biblatex}
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@book{one,
author = {Guðbjörg Gunnarsdóttir},
title = {One},
date = {1980},
}
@book{two,
author = {Rakel Hönnudóttir and Harpa Þorsteinsdóttir and Anna Björk Kristjánsdóttir},
title = {Two},
date = {1982},
}
@book{three,
author = {Rakel Hönnudóttir and Harpa Þorsteinsdóttir and Elísa Viðarsdóttir},
title = {Three},
date = {1983},
}
@book{four,
author = {Katrín Jakobsdóttir},
title = {Four},
date = {1984},
}
@book{five,
author = {Jóhanna Sigurðardóttir},
title = {Five},
date = {1985},
}
\end{filecontents}
\addbibresource{\jobname.bib}
\DeclareNameAlias{default}{given-family}
\DeclareNameAlias{sortname}{default}
\DeclareNameAlias{labelname}{default}
\DeclareSortingNamekeyTemplate{
\keypart{
\namepart{given}
}
\keypart{
\namepart{prefix}
}
\keypart{
\namepart{family}
}
\keypart{
\namepart{suffix}
}
}
\begin{document}
\cite{one}
\cite{two}
\cite{three}
\cite{four}
\cite{five}
\printbibliography
\end{document}
The Ethiopian naming system is similar to the Icelandic system. See Bibtex/Biber: how to cite an author using Ethiopian conventions? for an implementation that deals with Ethiopian names.
The input format is more tedious, but this solution implements a uniqueness feature of Icelandic names.
\documentclass[icelandic]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{babel}
\usepackage{csquotes}
\begin{filecontents*}{icelandic-names.dbx}
\DeclareDatamodelConstant[type=list]{nameparts}{prefix,family,suffix,given,middle,patronymic,papponymic}
\end{filecontents*}
\usepackage[style=authoryear,datamodel=icelandic-names]{biblatex}
\DeclareSortingNamekeyTemplate[icelandic]{
\keypart{
\namepart{given}
}
\keypart{
\namepart{patronymic}
}
\keypart{
\namepart{papponymic}
}
}
\DeclareUniquenameTemplate[icelandic]{
\namepart[base=true]{given}
\namepart[disambiguation=full]{patronymic}
\namepart[disambiguation=full]{papponymic}
}
\DeclareNamehashTemplate[icelandic]{
\namepart[hashscope=full]{given}
\namepart[hashscope=full]{patronymic}
\namepart[hashscope=full]{papponymic}
}
\DeclareNameAlias{sortname}{family-given}
\DeclareNameFormat{given-family}{%
\ifsortingnamekeytemplatename{icelandic}
{\usebibmacro{name:icelandic}
{\namepartgiven}
{\namepartpatronymic}
{\namepartpapponymic}}
{\ifgiveninits
{\usebibmacro{name:given-family}
{\namepartfamily}
{\namepartgiveni}
{\namepartprefix}
{\namepartsuffix}}
{\usebibmacro{name:given-family}
{\namepartfamily}
{\namepartgiven}
{\namepartprefix}
{\namepartsuffix}}}%
\usebibmacro{name:andothers}}
\DeclareNameFormat{family-given}{%
\ifsortingnamekeytemplatename{icelandic}
{\usebibmacro{name:icelandic}
{\namepartgiven}
{\namepartpatronymic}
{\namepartpapponymic}}
{\ifgiveninits
{\usebibmacro{name:family-given}
{\namepartfamily}
{\namepartgiveni}
{\namepartprefix}
{\namepartsuffix}}
{\usebibmacro{name:family-given}
{\namepartfamily}
{\namepartgiven}
{\namepartprefix}
{\namepartsuffix}}}%
\usebibmacro{name:andothers}}
\DeclareNameFormat{labelname}{%
\ifuniquenametemplatename{icelandic}
{\usebibmacro{labelname:icelandic}}
{\usebibmacro{labelname:western}}%
\usebibmacro{name:andothers}}
\newbibmacro{labelname:icelandic}{%
\iffieldequalstr{uniquepart}{base}
{\usebibmacro{name:icelandic}
{\namepartgiven}
{\empty}
{\empty}}
{\iffieldequalstr{uniquepart}{patronymic}
{\usebibmacro{name:icelandic}
{\namepartgiven}
{\namepartpatronymic}
{\empty}}
{\usebibmacro{name:icelandic}
{\namepartgiven}
{\namepartpatronymic}
{\namepartpapponymic}}}}
\newbibmacro{labelname:western}{%
\ifcase\value{uniquename}%
\usebibmacro{name:family}
{\namepartfamily}
{\namepartgiven}
{\namepartprefix}
{\namepartsuffix}%
\or
\ifuseprefix
{\usebibmacro{name:given-family}
{\namepartfamily}
{\namepartgiveni}
{\namepartprefix}
{\namepartsuffixi}}
{\usebibmacro{name:given-family}
{\namepartfamily}
{\namepartgiveni}
{\namepartprefixi}
{\namepartsuffixi}}%
\or
\usebibmacro{name:given-family}
{\namepartfamily}
{\namepartgiven}
{\namepartprefix}
{\namepartsuffix}%
\fi}
\newbibmacro*{name:icelandic}[3]{%
\usebibmacro{name:delim}{#1#2#3}%
\usebibmacro{name:hook}{#1#2#3}%
\mkbibcompleteice{%
\mkbibicegiven{#1}%
\ifdefvoid{#2}{}{\bibnamedelimd\mkbibicepat{#2}\isdot}%
\ifdefvoid{#3}{}{\bibnamedelimd\mkbibicepap{#3}\isdot}}}
\let\mkbibcompleteice\mkbibcompletename
\let\mkbibicegiven\mkbibnamefamily
\let\mkbibicepat\mkbibnamegiven
\let\mkbibicepap\mkbibnamegiven
\begin{filecontents*}{\jobname.bib}
@book{one,
author = {given=Guðbjörg, patronymic=Gunnarsdóttir, nametemplates=icelandic},
title = {One},
date = {1983}
}
@book{four,
author = {given=Katrín, patronymic=Jakobsdóttir, nametemplates=icelandic},
title = {Four},
date = {1984},
}
@book{five,
author = {given=Jóhanna, patronymic=Sigurðardóttir, nametemplates=icelandic},
title = {Five},
date = {1985},
}
@book{six,
author = {given=Guðrún Eva, patronymic=Mínervudóttir, nametemplates=icelandic},
title = {Six},
date = {1976},
}
@book{seven,
author = {given=Vilborg, patronymic=Dagbjartsdóttir, nametemplates=icelandic},
title = {Seven},
date = {1987},
}
@book{eight,
author = {given=Katrín, patronymic=Jónsdóttir, nametemplates=icelandic},
title = {Eight},
date = {1988},
}
\end{filecontents*}
\addbibresource{\jobname.bib}
\addbibresource{biblatex-examples.bib}
\begin{document}
\textcite{one,sigfridsson,knuth:ct:a,knuth:ct:b,knuth:ct:c,cicero,geer,five,six,seven}
\textcite{five}
\textcite{four}
\textcite{eight}
\printbibliography
\end{document}
Theoretically the logic could be inverted if you only use given and partonymic for icelandic names, so that all names are parsed as Icelandic unless specified otherwise.
