Using different footnote numbering styles at the same time

Try it with the manyfoot package

usepackage[para]{manyfoot}% allow [para] option

DeclareNewFootnote[para]{A}[alph] % define footnoteA
%   for alphabetic paragraph footnotes

Oops, the question asked for alphabetic ordinary footnotes, not alphabetic paragraph footnotes. I don't see that manyfoot provides any mechanism for restyling footnotes. So either declare both sets of footnotes as "new" variants

DeclareNewFootnote{A}[alph] % define footnoteA = alphabetic
DeclareNewFootnote[para]{N} % and footnoteN = numeric in paragraph

or declare just the new alphabetic series while changing the numbering of regular footnotes

renewcommandthefootnote{alph{footnote}} % footnote uses letters
DeclareNewFootnote[para]{N} % and footnoteN = numeric in paragraph

I initially recommend the bigfoot package since it fixes up and extends the manyfoot package. But it seems to not fill the line in para mode; I only see two notes on a line.

usepackage{bigfoot}
DeclareNewFootnote{default}[alph]% default footnote alphabetic
DeclareNewFootnote[para]{P} %  footnoteP for numeric

A point in favor of bigfoot is the ability to apply declarations to the ordinary footnotes, called "default".

The memoir class (a superset of the book, report and article classes) provides a variety of footnote styles, layouts, and numbering. For your case try this:

% footnoteprob.tex  SE 576072

documentclass{memoir}

paragraphfootnotes
newfootnoteseries{X}
renewcommand{thefootnoteX}{alph{footnoteX}}


begin{document}

This is a testfootnoteX{This word ...} to show how to createfootnote{One ...}
two different kinds of footnotes with different numbering 
stylesfootnote{The ...} at the samefootnote{Same} 
timefootnoteX{Two special ...}.

end{document}