Ways to speed up numerical integration

Let NDSolve do the job one time (instead of Integrate many times)

Edit Make it faster. AbsouteTiming for all calculations == 0.015 seconds.

(tpw[t_] = Total@pulses;

 gsol = g /. 
   First@NDSolve[{g'[t] == tpw[t], g[0] == 0}, g, {t, 0, 250}];

 norm = gsol[250] ;(*11.*)

{t05 = t /. First@FindRoot[gsol[t] == .05 norm, {t, 0, 150}],
 t95 = t /. First@FindRoot[gsol[t] == .95 norm, {t, 0, 150}]}
) //  AbsoluteTiming

(*   {0.0155646, {4.06347, 128.699}}   *)

NIntegrate[Total@pulses, {t, 0, t05}]/norm
(*   0.05   *)

NIntegrate[Total@pulses, {t, 0, t95}]/norm
(*   0.95   *)

Plot[{1, .05, .95, gsol[t]/norm}, {t, 0, 150}]

Should be a bit faster... Long story short: Supply the Jacobian of your equation to FindRoot, too.

ϕ = t \[Function] Evaluate[Total@pulses];
ClearAll[Φ];
Φ[x_?NumericQ] :=    NIntegrate[ϕ[t], {t, 0., x}, PrecisionGoal -> 8];

rhs = 0.05 Φ[tmax];

(*An approximate inverse of Φ*)

tlist = Subdivide[0., tmax, 100];
Ψ = Quiet[
   Interpolation[
     Transpose[{Accumulate[N[ϕ /@ tlist]], tlist}],
     InterpolationOrder -> 1]
   ];
(*Initial guess:*)
x0 = Ψ[rhs];

(*Using Newton's method.The Jacobian of Φ is easy enough to compute!;)*)

sol = FindRoot[Φ[x] == rhs , {x, x0}, 
     Jacobian :> {{ϕ[x]}}]; // AbsoluteTiming // First

0.056017

Test:

Φ[x] - rhs /. sol

-1.11022*10^-16

Btw.: There is no point in enforcing 100(!) digits of precision with PrecisionGoal -> 100 if FindRoot's default PrecisionGoal is used (it's about 8, isn't it?).