Order of the variables in a polynomial?

One more thing... would it be possible to remove the powers of 1 (stuff like $3^1$ to be just $3$)? Or if not, at least to remove $1^1$ completely? Thank you!

Perhaps there was too much nesting. Try this variation with fo:

With[{order = "Lexicographic"}, 
 StringJoin[
  Riffle[Table[
    poly = MonomialList[K[Sqrt[#]/Tanh[Sqrt[#]] &, i] /. c -> p, 
      Table[Subscript[p, k], {k, 1, 7}], order];
    gcd = GCD @@ poly /. Rational[n_, d_]*c_ :> d;
    ToString[
     Inactive[Set][Subscript[L, i], 
      DisplayForm@
       RowBox[{1/primeFactorForm[gcd], "(", 
         RowBox[Flatten[(List @@ 
               Distribute[gcd*poly]) /. {Times[Rational[n_, d_], 
                e__] :> {If[n/d > 0, "+", "-"], 
                primeFactorForm[Abs@n]/primeFactorForm[Abs@d], e}, 
              Times[n_Integer, e__] :> {If[n > 0, "+", "-"], 
                primeFactorForm[Abs@n], e}}] /. {"+", a___} :> {a}], 
         ")"}]], TeXForm],
    {i, 3, 7}],
   "\\\\"]]]

Over my 25+ years of usage I have learned a fair number of tricks. Unfortunately, the precise formatting of expressions is not easy in Mathematica. As a gateway into the subject, I would suggest the documentation on RowBox. Don't forget to wrap everything into DisplayForm. An occasional peek into CellExpression may give clues on low-level box structures.

With[{order = "Lexicographic"}, 
 StringJoin[
  Riffle[Table[
    poly = MonomialList[K[Sqrt[#]/Tanh[Sqrt[#]] &, i] /. c -> p, 
      Table[Subscript[p, k], {k, 1, 7}], order]; 
    gcd = GCD @@ poly /. Rational[n_, d_]*c_ :> d;
    ToString[
     Inactive[Set][Subscript[L, i], 
      DisplayForm@
       RowBox[{1/primeFactorForm[gcd], "(", 
         RowBox[(List @@ 
              Distribute[gcd*poly]) /. {Times[Rational[n_, d_], 
               e__] :> 
              RowBox[{If[n/d > 0, "+", "-"], 
                primeFactorForm[Abs@n]/primeFactorForm[Abs@d], e}], 
             Times[n_Integer, e__] :> 
              RowBox[{If[n > 0, "+", "-"], primeFactorForm[Abs@n], 
                e}]} /. {"+", a___} :> {a}], ")"}]], TeXForm], {i, 3, 
     7}], "\\\\"]]]

There was a Plus that rearranged everything. Try this:

Manipulate[
 Table[
  poly = MonomialList[K[Sqrt[#]/Tanh[Sqrt[#]] &, i] /. c -> p, 
    Table[Subscript[p, k], {k, 1, 7}], order]; 
  gcd = GCD @@ poly /. Rational[n_, d_]*c_ :> d;
  DisplayForm@
   RowBox[{1/primeFactorForm[gcd], "(", 
     RowBox[(List @@ 
          Distribute[gcd*poly]) /. {Times[Rational[n_, d_], e__] :> 
          RowBox[{If[n/d > 0, "+", "-"], 
            primeFactorForm[Abs@n]/primeFactorForm[Abs@d], e}],
         Times[n_Integer, e__] :> 
          RowBox[{If[n > 0, "+", "-"], primeFactorForm[Abs@n], 
            e}]} /. {"+", a___} :> {a}], ")"}], {i, 3, 7}],
 {order, {"Lexicographic", "DegreeLexicographic", 
   "DegreeReverseLexicographic", "NegativeLexicographic", 
   "NegativeDegreeLexicographic", 
   "NegativeDegreeReverseLexicographic"}}]

You can give your ordering to the polynomials by using MonomialList instead of List@@:

poly = MonomialList[K[Sqrt[#]/Tanh[Sqrt[#]] &, i] /. c -> p, 
  Table[Subscript[p, i], {i, 1, 7}], "NegativeLexicographic"];
gcd = GCD @@ poly /. Rational[n_, d_]*c_ :> d;

For example here I have inserted the "NegativeLexicographic" ordering, but there are many others. There is a tutorial on PolynomialOrderings in the documentation.