Current source as load in a differential amplifier
Your initial guess is right: you size M3 and M4 so that with zero differential input the current through the two branches is the same, i.e. Iss/2.

For small differential inputs this condition is always nearly true, i.e. the current through M1 and M2 drains is nearly Iss/2 if you neglect a very tiny difference. Luckily enough this very tiny difference flows in the output resistance of M3 and M4, that is very high, thus producing an high output voltage swing and (hopefully) your required gain.

When you apply a big signal, such as what you describe in the last part of your question, many things happen. The most important is that your previous models does not work any more: you just can't treat M3 and M4 as ideal current sources because some hypothesis you made are no longer true.

Let's say you apply the maximum input voltage: M1 is fully on and M2 is fully off. Let's also assume that Iss is ideal. All the current must flow through the left branch, and none can flow in the right since M2 is off. M3 will accomodate for the higher current with an higher Vds, that's where the assumption M3 and M4 carry the same current fails: their Vds is now quite different.

Since for M3 you know Vgs and Ids the Vds is known too -> you can calculate the output voltage. What you can't know is M1 Vds, together with its Vgs. What happens here is that the ideal current source provides the necessary drain voltage to keep the circuit standing on its feet, possibly a voltage below ground.

If that is not an ideal current source, but a mosfet, say M5, then the whole story is a battle between M5 and M3: M5 drain voltage (i.e. M1 source voltage) will start to go down, it would really like to go even below ground, but unfortunately you probably need some saturation voltage on that node (~200mV), below that M5 just stops behaving like a current source and its drain current drops, and you finally get zero current on the right branch, and a current that's between Iss/2 and Iss on the left branch.