Doping and inter plane coupling for cuprates

First question

1) If we increase the doping, how does the hopping strength t and U change in the Mott insulator picture? Is the effective spin hamiltonian still valid in certain 2D regions? My naive picture is wherever there is doping atom, there is no such interaction near that doped region. But I could imagine, t and U change completely and our perturbation theory for AFM does'nt work anymore.

parameters and filling

There are two different concepts: model and filling. First of all, what you said about $t$ and $U$ is "Hubbard model", $$H=-t\sum_{i,j}c_i^\dagger c_j+h.c.+\sum_i Un_{i,\uparrow}n_{i,\downarrow}$$ which describes interacting electrons with Hilbert space for each site: empty, single occupation, double occupation. And $U$ is just the energy cost for double occupation. Most important point is such model can describe system both strong coupling (large-U) and weak coupling (large-U), both half-filling and doping. And the choosing parameters and filling is independent. In the other words, we can let the system keep strong coupling(large $U$ and small $t$), but changes its filling (e.g. from half-filling to only single electron).

Effective model

However, the independent of filling and parameters is correct just for the Hubbard model, including both spin fluctuation and charge fluctuation. The standard paradigm in condensed matter theory is to derive low-energy effective model since most of time we only care about low-temperature physics, i.e. high-energy states is hard to be excited at low-temperature, so that we can project out these high-energy states to obtain an "effective model" with smaller Hilbert space, which can only describe the low-energy physics. However, the projection depends on both parameters and filling. In details, if we only consider the strong coupling (i.e. cuprates are actually the typical case for strong coupling, large-$U$), the high-energy states are those with double occupation due to large-$U$ cost so that we need to project them. For half-filling, such projection means every electrons should stay at its sites and can not hopping since hopping will always connect single occupation with double occupation for hal-filling, which is so-called "Mott insulator". For this phase(actually the parent of cuprates), systems only contains spin interaction, thus we can write an effective model including only spin-exchange interaction: $$H=J \sum_{<i j>} S_{i} \cdot S_{j}$$ But as you said, when we dope the "Mott insulator", the hopping electrons/holes actually can hop so that we need to add back the kinetic term, i.e. $-t\sum_{i,j}c_i^\dagger c_i +h.c.$ . However, such process is not trivial since the projection also introduce the non-double occupation constraints, so that the resulting effective Hamiltonian now is: $$H=P_s(-t\sum_{i,j}c_i^\dagger c_i) P_s+h.c.+J \sum_{<i j>} S_{i} \cdot S_{j}$$ where $P_s$ is non-double occupation constraints: $$n_{i\uparrow}+n_{i\downarrow}\leq1$$

but after all, when you dope the Mott insulator, the kinetic part(change fluctuation) will be partially restore. Also, the parameters $J$ depends on both $t$ and $U$: $$J=-\frac{t^2}{U}$$ This new model is called "t-J" model, if you are interested in the details for projection from Hubbard model to t-J model, Ch.3. Auerbach, Interacting electrons and quantum magnetism is a good reference.

So, answer your first question in one sentence: when you increase dopping, $t$ and $U$ don't change, spin-exchange interaction remains but there also exist an additional partially hopping terms, where the projection in this term is highly affected by the number of dope.

Supplement

The goal of non-double occupation is to project out the high-energy sector. But this constraint is just apparent for hole-doping system (i.e. number of electrons smaller than half-filling) since we know the high-energy sector is just the states with double-occupation:

But for electron-doping system, this picture and the form of constraint seems strange since there always exists double-occupation states. To unify this problem, it is important to note that the "high-energy" is relative, and the "lowest" states for such system is: first add one electron in each site, then put the additional electrons in the double-occupation states. In the other words, there exists at least one electron in each site. Thus, the high-energy sector is the states which contains at least one site without any electrons(at least one empty site). Now, the constraint can be considered as : $$n_{i\uparrow}+n_{i\downarrow}\geq1$$

Second Question

Is there any coupling between the planes? (I know it seems two electrons are out of option for bonding. But maybe there are other effective and weaker coupling?)

Yes, there exists some other coupling, like phonon. And, there also exist spin-exchange interaction between layers.