How to make an arrow from a node to itself have a nice arc?

1. I usually oput a loop argument to the line for the arrow. It is not perfect arc-like, but reasonable.
2. [above=10mm]
3. shorten is the way to go. It does not clip the arrow tip. I put it to 1pt on all arriving edges but you can set to whatever you like. On bended lines like yours it looks strange if you take high values (like 1cm).
4. Draw the line from the node and out.

Code:

\documentclass[11pt]{article}
\usepackage{tikz}
\usetikzlibrary{arrows,shapes,snakes,automata,backgrounds,petri}
\begin{document}

\begin{tikzpicture}[node distance=1.7cm]
\tikzstyle{place}=[circle,thick,draw=gray!75,fill=gray!20,minimum size=6mm]

\begin{scope}
\node [place] (s1c) {$s_1$} node[above=10mm]{$c<0$};
\node [place, below of=s1c] (s2c) {$s_2$};

\draw[thick,->,shorten >=1pt] (s2c) to [out=135,in=225] (s1c);
\draw[thick,-|,shorten >=1pt] (s1c) to [out=-45,in=45] (s2c);

\draw[thick,->,shorten >=1pt] (s1c) to [out=90,in=180,loop,looseness=4.8] (s1c);
\draw[thick,-|,shorten >=1pt] (s2c) to [out=-90,in=0,loop,looseness=4.8] (s2c);

\draw[thick,<-,shorten <=1pt] (s1c) -- +(45:1cm);
\end{scope}

\end{tikzpicture}

\end{document}

EDIT

One (not too elegant) way to get nicer arcs for the self loops is to use an arc. It can probably be better derived, here I used a more iterative (trial-and-error) approach. Replace the arcs with

\draw[red,thick,->] (s1c.90) arc (0:264:4mm);
\draw[red,thick,-|] (s2c.-90) arc (180:180+264:4mm);

to get

Late edit

To formulate a command for a circled arc one can do e.g.

\documentclass[border=5mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{calc,arrows.meta}
%%
\usepackage{xparse}
\NewDocumentCommand\Cycle{O{} m m m O{} m}{%
  % [opt arg cycle]{Node}{Angle}{Node size}[opt arg arch node]{cycle size}
  \draw[#1](#2.{#3+asin(#6/(#4*1.41))}) arc (180+#3-45:180+#3-45-270:#6/2) #5;
}
%%
\begin{document}
\begin{tikzpicture}
  \node[draw,circle,minimum width=20mm,inner sep=0pt,thick](A) at (0,0) {A};
  \Cycle{A}{0}{20mm}{10mm}
  \Cycle[-{Latex[scale=1.2]}]{A}{90}{20mm}{15mm}
  \Cycle[{Latex[scale=0.8]}-]{A}{90}{20mm}{20mm}
  \Cycle[red,dashed,->]{A}{180}{20mm}[{node[anchor=0,pos=0.5,blue]{Test}}]{10mm}
  \foreach \r in {2,4,...,20}{%
    \Cycle[-{Latex[scale=0.5]}]{A}{270}{20mm}{\r mm}
  }
\end{tikzpicture}
\end{document}

With looseness option and use node names from automata library:

Edit: it seems that I overlooked latter added sub questions. According to them is corrected first example below. The code should be self-explanatory ...

\documentclass[11pt]{article}
\usepackage{tikz}
\usetikzlibrary{arrows, automata, calc, positioning}

\begin{document}
    \begin{tikzpicture}[
shorten >=1pt,>=stealth,
     node distance = 2cm, on grid,
every state/.style = {draw=gray!75, fill=gray!20, thick},
        bend angle = 45
                        ]
\node [state] (s1c) [label={[yshift=6mm] $c<0$}] {$s_1$};
\node [state] (s2c) [below=of s1c] {$s_2$};
%
\draw[thick,->] (s2c) edge [bend right] (s1c)
                (s1c) edge [looseness=5, out= 90, in=180] (s1c)
                ($(s1c)+(45:13mm)$) to (s1c);
\draw[thick,-|] (s1c) edge [bend right] (s2c)
                (s2c) edge [looseness=5, out=270, in=  0] (s2c);

    \end{tikzpicture}
\end{document}

It was not clear, if $c<0$ is condition for the loop at node s_1 or belong to node or image. In the former it should be label of the loop:

 \draw[thick,->] (s1c) to [looseness=5, out= 90, in=180] node[left] {$c<0$} (s1c);

and deleted as label of the first node. In this case you will obtain:

This answers question 1 and implicitly questions 2 and 4 (just use the above of (or above right of) syntax used elsewhere in this code). You already know how to take care of 3.

\documentclass[11pt]{article}
\usepackage{tikz}
\usetikzlibrary{arrows,shapes,snakes,automata,backgrounds,petri}
\usetikzlibrary{positioning}

\begin{document}

    \begin{tikzpicture}[node distance=1.7cm]
    \tikzstyle{place}=[circle,thick,draw=gray!75,fill=gray!20,minimum size=6mm]

    \begin{scope}
    \node [place] (s1c) [label=above right:$c<0$] {$s_1$};
    \node [place] (s2c) [below of=s1c] {$s_2$};
    \node[above = 1cm of s1c] (poop){};
    \node[left = 1cm of s1c] (peep){};
    \node[below = 1cm of s2c] (holy){};
    \node[right = 1cm of s2c] (moly){};

    \draw[thick,->] (s2c) to [out=135,in=225] (s1c);
    \draw[thick,-|] (s1c) to [out=-45,in=45] (s2c);

    \draw[thick,->] (s1c.north) ..  controls (poop) and (peep) .. (s1c.west);
    \draw[thick,->] (s2c.south) ..  controls (holy) and (moly) .. (s2c.east);
    \end{scope}

    \end{tikzpicture}

\end{document}