Would like input and output printed on same line, w/o needing extra syntax

You can fix up Partial Solution 1 with your additional requirements with

$PrePrint = (TraditionalForm@HoldForm[In[line] = #] /. line -> $Line /. DownValues[In] /.
     {HoldPattern[a_ = a_] :> a,
      HoldPattern[a_ = HoldForm[a_]] :> a,
      HoldPattern[(c : (a_ = b_)) = b_] :> c,
      HoldPattern[(a_ = b_) = c_] :> HoldForm[a = b = c]
      }
    ) &;

$Post = (Replace[#, Null -> HoldForm[In[line]] /. line -> $Line /. DownValues[In]]) &;

The first three additional replacement rules in $PrePrint address redundancy in the output, the fourth replacement rule removes the unwanted parentheses, and the $Post command makes it so there is always output, even if there normally would be none.

Now you have

Integrate[x^2, x]

$\int x^2 \, dx=\frac{x^3}{3}$

int = Integrate[x^2, x]

$\text{int}=\int x^2 \, dx=\frac{x^3}{3}$

int/x

$\frac{\text{int}}{x}=\frac{x^2}{3}$

%/x

$\frac{\%}{x}=\frac{x}{3}$

a = 2

$a=2$

g[x_] := Sin[x]

$g(x\_):=\sin(x)$

g[Pi]

$g(\pi)=0$

Note that (a = b) = c has a different meaning than a = b = c -- the latter first assigns c to b, and then assigns b to a, while the former first assigns b to a, and then c to a. That's why the parentheses are put there in the first place. But that should only be an issue for you if you try to copy and paste and then re-evaluate the input.

Some explanations of the code:

Mathematica stores all of the previously entered input in the function In, as values for In[1], In[2], etc. You can get a list of all of these values as a list of replacement rules with Downvalues[In]. Try evaluating Downvalues[In] by itself (after doing some other calculations) to see. So Downvalues[In] already has the correct syntax to be used in a replacement operation.

$PrePrint and $Post are similar -- they both apply a function to every output expression. The main difference between them is that $Post is applied before the value of the output is assigned to the Out variable, and $Preprint is applied after. You can see this in action with (in a freshly started kernel):

$PrePrint = f

f[f] 

a

f[a]

$Post = g

f[g[g]]

a

f[g[a]]

$PrePrint =.

g[Null]

$Post =.

DownValues[Out]

{HoldPattern[%1] :> f, HoldPattern[%2] :> a, HoldPattern[%3] :> g[g], 
 HoldPattern[%4] :> g[a], HoldPattern[%5] :> g[Null], HoldPattern[%6] :> Null}

Pondering on that sequence of input and output should help you understand $PrePrint and $Post. Note that $Post is applied before $PrePrint, and the effects of $Post show up in the downvalues for Out, while the effects of $PrePrint do not.

This solution does most of what your are looking for, producing stylized output much like what @Simon Rochester's answer shows. Additionally, this writes to a new notebook window:

$Pre=.; 

If[Notebooks["Output"]=={}, nb= CreateDocument[ Null, {
        WindowSize->Medium, WindowTitle->"Output", WindowMargins->Automatic, 
        StyleDefinitions-> FrontEnd`FileName[{"Report"},"StandardReport.nb",CharacterEncoding->"UTF-8"]
    }] ]; 

Clear[fPrint]; 
SetAttributes[fPrint, HoldAll]; 

fPrint[expr_]:= Block[ { exprPrint, exprHold=HoldForm[expr] },
    NotebookWrite[nb, SelectionMove[nb, Next, Cell]
    ; CellGroupData[
        {
        Cell[BoxData[ToBoxes[HoldForm[expr], StandardForm]], "Input"], 
            ( 
             exprPrint = If[
                MatchQ[exprHold, HoldForm[Set[x_,y_]]/;AtomQ[y]]
                , exprHold
                , exprHold == expr ]  
             ; Cell[BoxData[ToBoxes[exprPrint, TraditionalForm]], "Output", Editable->True] 
            )
        }
        , Open], AutoScroll-> False] 
        ; expr
    ] // Quiet;

$Pre = fPrint;