This paper seems relevant to your question. If I'm reading the abstract correctly, the answers to your questions are:
Q: It seems that if the coherence length of a laser is big enough, it is possible to observe a (moving) interference picture by combining them. Is it true?
A: Yes
Q: How fast should photo-detectors be for observing of the interference of beams from two of the "best available" lasers?
A: 1 millisecond or faster
Q: What is the coherence length of the best-available laser?
A: More than 300 km
Q: More specifically, does there exist any (optical single-wavelength) laser with coherence length exceeding 500 meters?
A: Yes
The abstract in the paper:
Interference fringes produced by a pair of intracavity stabilized diode laser beams, each impinging separately on one aperture of a double slit, are recorded on a linear charge-coupled device array. The peculiar result of the experiment is that the fringe system is found to persist for a time of the order of 1 ms and loses contrast for longer integration times. This implies that the individual linewidths of the two beams from the stabilized lasers are narrower than 1 kHz and that the average drift rates of the central peaks are far less than 0.1 MHz/s. The device was built within the advanced undergraduate electronics laboratory of the department of physics and represents a considerable improvement over previous demonstration apparatuses used to detect interference fringes from independent lasers.
An interesting 1986 review of interference from independent sources:
"Interference between independent photons", Rev. Mod. Phys. 58, 209–231 (1986)
When you first apply current to a laser diode, it does behave as an LED. Light is output across a (relatively) broad spectrum by spontaneous emission.
But once the current reaches the threshold current, then positive feedback causes one (or a few) modes to oscillate. Further increases in input power will increase the ouput in those particular modes, but the gain will be pinned at the threshold level. Since gain is mainly a function of carrier density, this means the carrier density is also pinned, and spontaneous emission won't increase.
Similar gain-pinning mechanisms are seen in most laser systems.
Best Answer
For questions like this I always recommend finding the relevant article on Sam's Laser FAQ, which is an incredible practical resource. However,The short answer is that they are much better than standard diode lasers on all three points.
The emitted beam from an Nd:YAG or Nd:YVO crystal is very pure 1064 nm light and the nonlinear crystal, typically potassium titanyl phosphate (or KTP for short), doubles it precisely to 532 nm ($\pm1$ nm I'd guess). You can find several diagrams of the internals of green laser pointers in the link above. My guess is that main effect of variations in the performance of the 808 nm pump diode would be to reduce the coupling efficiency and thus the output power, but I believe that Nd:YVO is relatively tolerant to such variations.
Hazarding a wild guess based on your question, I'd say that you might be interested in doing some holography experiments with a green laser pointer. This paper details how to do professional quality holography with a green laser pointer as the source.
Apologies for not providing quantitative answers to your question, but hopefully I've provided the practical info that you need.