For the function line issue, it’s because your functions are defined from 9.5 (or 10.5 as you would like) to 200 (even though you are only plotting up to 37), and by default ROOT draws TF1 by sampling 100 points in x, so there aren’t enough points; you can change that number with TF1::SetNpx:
https://root.cern/doc/master/classTF1.html#a6a04facaa58306988a3189b4e8713aa5
For your case, you could increase that npx (around 500 seems to look ok), or reduce the upper limit of your functions (e.g. to 40), or a combination of both. Using DrawFrame as @couet suggested, and these 2 options:
TCanvas *c1 = new TCanvas("c1","",700,500);
gStyle->SetOptStat(0);
gStyle->SetLegendBorderSize(0);
c1->DrawFrame(0,0,37,1.30);
TGaxis *axis = new TGaxis(37,0.0, 37, 1.15,0,1.5,51,"+L");
axis->SetNdivisions(505);
axis->SetLabelFont(gStyle->GetLabelFont("Y"));
axis->SetTitleFont(gStyle->GetTitleFont("Y"));
axis->SetTitle(" Background Rate [MHz]");
axis->CenterTitle();
axis->SetTitleSize(0.08);
axis->SetTitleOffset(0.5);
axis->Draw();
TLine *line = new TLine(10.5,0,10.5,1.30);
line->SetLineStyle(2);
line->Draw();
double Act_Zr88_0 = 1.11; // GBq
double tHalf_Zr88 = 83.4; // days
double tHalf_Y88 = 106.629; // days
double Eff_Zr88 = 0.0002;
double Eff_Y88 = 0.0035;
// BLUE, increasing Npx
TF1 *fBkg_Zr88 = new TF1("fBkg_Zr88","1000*[3]*[0]*exp(-log(2.0)*x/[1])",10.5,200);
fBkg_Zr88->SetParameters(Act_Zr88_0, tHalf_Zr88, tHalf_Y88, Eff_Zr88);
fBkg_Zr88->SetLineColor(kBlue+2);
fBkg_Zr88->SetLineWidth(4);
fBkg_Zr88->SetLineStyle(7);
fBkg_Zr88->SetNpx(500);
fBkg_Zr88->Draw("same");
// RED, reducing range of TF1
TF1 *fBkg_Y88 = new TF1("fBkg_Y88","1000*[4]*(1.0/[1])/(1.0/[2]-1.0/[1])*[0]*(exp(-log(2.0)*x/[1])-exp(-log(2.0)*x/[2]))",10.5,40);
fBkg_Y88->SetParameters(Act_Zr88_0, tHalf_Zr88, tHalf_Y88, Eff_Zr88, Eff_Y88);
fBkg_Y88->SetLineColor(kRed+2);
fBkg_Y88->SetLineWidth(4);
fBkg_Y88->SetLineStyle(7);
fBkg_Y88->Draw("same");
