How fast do the magnetic domains align with the external field? Let say we bring a normal (not super strong) magnet near sort of an unmagnetized iron block.
1) Do the magnetic domains align instantly? Does it make any difference on magnetization strength if we keep the magnet there for a second or for a minute?
2) If we do not use a strong enough external field, not all the domains align with the field. Am I right on this?
 A: Sorry in advance for a brief answer: I am writing it fast and from the top of my head.
In general "no" and "no". 
First answer is always "no". Have you heard of hysteresis? Look it up. The magnetization time depends on the type of ferromagnet (diamagnet) and on the strength of magnetic field (also temperature etc., but that is secondary). Second question is always definite "no" quantum mechanically. Magnetization is alignment of spin domains with respect to external magnetic field... Spins... Does spin vector have a definite alignment direction? Classically, there are cases when at max magnetization you can assume complete alignment.
However, if your interest is limited by some general physics course, you might be dealing with models, where approximation of instantaneous magnetization can be applicable and where domain walls-related effects are neglected. Then your answers are "yes" and "yes".
For the future: please, specify theoretical framework you are working in or at least cite sources that you are using. Otherwise the answers will always be broad and fairly useless...
A: *

*Answer is no. Alignment begins quickly. Nothing is instantaneous, but some magnetization begins to build right away. However the time scale to reach the ultimate equilibrium can be very long with ferromagnets; years even. Consider, for example, old-fashioned cans of food sitting in the larder. They would slowly get magnetized by earth's field over a period of years.

*I can't help thinking that you won't get full alignment at a non-zero temperature. For a ferromagnet this may be a very small effect (because the effective internal field is large), but I think the alignment is not quite complete at any temperature except absolute zero. At non-zero temperature, there is a competition between thermal fluctuation which tends to randomize the magnetization and internal + applied field which tends to increase the magnetization. In consequence, low applied field gives strong but not complete magnetization; strong applied field gives somewhat stronger magnetization.
