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lpp3d1 = ListPointPlot3D[{#[[3 ;; 5]]} & /@ Sidata1, 
   PlotStyle -> colorsSi, AxesLabel -> {"q", "s2", "α"}, 
   ImageSize -> Large, LabelStyle -> {18},  PlotLabel -> Style["Si", 24]];

###1.

Remove braces from {#[[3 ;; 5]]} & in the first argument of ListPointPlot3D:

ListPointPlot3D[#[[3 ;; 5]] & /@ Sidata1] /. Point -> Line

###2.

If you want the line retain colors associated with points, use a more elaborate replacement rule:

lpp3d1 /. points : {{Directive[_, _], Point[_]} ..} :> 
   {Thick, Line[points[[All, 2, 1, 1]], VertexColors -> points[[All, 1, 2]]]}

If you want to show both the points and the lines, use

lpp3d1 /. points : {{Directive[_, _], Point[_]} ..} :> 
   {points, Thick, Line[points[[All, 2, 1, 1]], VertexColors -> points[[All, 1, 2]]]}

lpp3d1 = ListPointPlot3D[{#[[3 ;; 5]]} & /@ Sidata1, 
   PlotStyle -> colorsSi, AxesLabel -> {"q", "s2", "α"}, ImageSize -> Large, 
   LabelStyle -> {18},  PlotLabel -> Style["Si", 24], ViewPoint -> {-3, -2, 1}];

###1.

Remove braces from {#[[3 ;; 5]]} & in the first argument of ListPointPlot3D and use replacement rule Point -> Line:

ListPointPlot3D[#[[3 ;; 5]] & /@ Sidata1, PlotStyle -> colorsSi, 
   AxesLabel -> {"q", "s2", "\[Alpha]"}, ImageSize -> Large, 
   LabelStyle -> {18}, PlotLabel -> Style["Si", 24], 
   ViewPoint -> {-3, -2, 1}] /. Point -> Line

###2.

If you want the line retain colors associated with points, use a more elaborate replacement rule:

lpp3d1 /. points : {{Directive[_, _], Point[_]} ..} :> 
   {Thick, Line[points[[All, 2, 1, 1]], VertexColors -> points[[All, 1, 2]]]}

If you want to show both the points and the lines, use

lpp3d1 /. points : {{Directive[_, _], Point[_]} ..} :> 
   {points, Thick, Line[points[[All, 2, 1, 1]], VertexColors -> points[[All, 1, 2]]]}

lpp3d1 = ListPointPlot3D[{#[[3 ;; 5]]} & /@ Sidata1, 
   PlotStyle -> colorsSi, AxesLabel -> {"q", "s2", "α"}, 
   ImageSize -> Full, LabelStyle -> {18},  PlotLabel -> Style["Si", 24]];

###1.

Remove braces from {#[[3 ;; 5]]} & in the first argument of ListPointPlot3D and combine the result with lpp3d1 using Show:

Show[lpp3d1, ListPointPlot3D[#[[3 ;; 5]] & /@ Sidata1] /. Point -> Line]

###2.

If you want the line retain colors associated with points, use a more elaborate replacement rule:

lpp3d1 /. points : {{Directive[_, _], Point[_]} ..} :> {points, 
   Thick, Line[points[[All, 2, 1, 1]], VertexColors -> points[[All, 1, 2]]]}

lpp3d1 = ListPointPlot3D[{#[[3 ;; 5]]} & /@ Sidata1, 
   PlotStyle -> colorsSi, AxesLabel -> {"q", "s2", "α"}, 
   ImageSize -> Large, LabelStyle -> {18},  PlotLabel -> Style["Si", 24]];

###1.

Remove braces from {#[[3 ;; 5]]} & in the first argument of ListPointPlot3D:

ListPointPlot3D[#[[3 ;; 5]] & /@ Sidata1] /. Point -> Line

###2.

If you want the line retain colors associated with points, use a more elaborate replacement rule:

lpp3d1 /. points : {{Directive[_, _], Point[_]} ..} :> 
   {Thick, Line[points[[All, 2, 1, 1]], VertexColors -> points[[All, 1, 2]]]}

If you want to show both the points and the lines, use

lpp3d1 /. points : {{Directive[_, _], Point[_]} ..} :> 
   {points, Thick, Line[points[[All, 2, 1, 1]], VertexColors -> points[[All, 1, 2]]]}

lpp3d1 = ListPointPlot3D[{#[[3 ;; 5]]} & /@ Sidata1, 
   PlotStyle -> colorsSi, AxesLabel -> {"q", "s2", "α"}, 
   ImageSize -> Full, LabelStyle -> {18},  PlotLabel -> Style["Si", 24]];
 
Show[lpp3d1, ListPointPlot3D[#[[3 ;; 5]] & /@ Sidata1] /. Point -> Line]

lpp3d1 = ListPointPlot3D[{#[[3 ;; 5]]} & /@ Sidata1, 
   PlotStyle -> colorsSi, AxesLabel -> {"q", "s2", "α"}, 
   ImageSize -> Full, LabelStyle -> {18},  PlotLabel -> Style["Si", 24]];

###1.

Remove braces from {#[[3 ;; 5]]} & in the first argument of ListPointPlot3D and combine the result with lpp3d1 using Show:

Show[lpp3d1, ListPointPlot3D[#[[3 ;; 5]] & /@ Sidata1] /. Point -> Line]

###2.

If you want the line retain colors associated with points, use a more elaborate replacement rule:

lpp3d1 /. points : {{Directive[_, _], Point[_]} ..} :> {points, 
   Thick, Line[points[[All, 2, 1, 1]], VertexColors -> points[[All, 1, 2]]]}