Electric charge of the Higgs field

Simply read the WP article and heed its consistency. It is in P&S conventions, so please do not look at Srednicki, whose opposite conventions evidently confuse you consistently. Now, $$Q=T_3+Y_w/2, $$ so for the Higgs doublet, Y =1, hence Q = +1 for the upper component and 0 for the lower component, the one that picks up the v.e.v., cf. (1) in the WP article. So both the physical Higgs and the goldston pumping into and out of the vacuum are neutral. SSB does not break charge.

The $\phi_1+i\phi_2$ has then charge + , whereas the individual real and imaginary components in it do not have a well-defined charge.

There are the corresponding terms in the lagrangian that involve the conjugate Higgs doublet, with co-equal goldstons $\phi_1-i\phi_2$ of negative charge, just as there are as many electrons as positrons, so to speak, poetically. Check all therms $\Phi ^* \cdot \Phi$ and their functions are neutral in charge, weak isospin and hypercharge.

It is a charge +1 goldston eaten by $(W_1-iW_2)/\sqrt{2}\equiv W^+$, with similar charge properties, perhaps counterintuitively: $W_1-iW_2$ ate $\partial (\phi_1+i\phi_2)$! I don't have a glib maxim for it, but it follows direct calculation from the covariantly completed Higgs kinetic terms: Remind yourself of how the resulting mass term emerges proportional to $W^+ W^-$, skipping Lorentz indices, so it conserves charge, likewise. A good SM text should help.

As always, check the electron-neutrino-Higgs Yukawa on the last line of (6) in the WP article to ensure you understand how it conserves charge, weak hypercharge, and weak isospin.

Besides, if two components of the Higgs field carry positive electric charge, the whole space (even the whole universe) is electrically positive since the Higgs field permeates the whole space. This is very doubtful and seems not reasonable to me. Is this case true?

You are confusing the fields of quantum field theory, with the particles generated by creation operators on these fields.

After all the whole space in QFT is covered by the electron field and the quark fields etc . these fields have no charges.