What are the best known results for the stable homotopy groups of spheres?

All your answers are somewhere in the green book!

In fact in Chapter 7 in the book Ravenel uses the Adams-Novikov spectral sequence to calculate the first thousand (stable) stems for $p=5$

The best method seems to be the Adams-Novikov spectral sequence with Brown-Peterson (co)homology.

The Lambda algebra can be found in Chapter 3 of the green book.

You could also try having a read of McCleary's 'A User's Guide to Spectral Sequences' - the classical Adams spectral sequence is treated in that (Chapter 9), and a brief introduction to the Adams-Novikov spectral sequence (Chapter 11)

Edit: One further comment. You don't really need the Lambda algebra to find a (minimal) free resolutions of $\mathbb{Z}/2\mathbb{Z}$ over the Steenrod algebra. Have a look in Mosher and Tangora's book, or in Allen Hatcher's spectral sequence's book

1) This Question is a little vague. You could be asking about how far out we know these stable homotopy groups and what the best up to date reference is. This is not the way people are approaching this problem currently. I believe Christian Nassau has the most extensive Adams Spectral Sequence computations. Is that what you are interested in?

2) The Adams Spectral Sequence is not easy to improve upon at the prime 2. However, at odd primes it is a result of Haynes Millers that the Adams-Novikov SS is a strict improvement over the Adams SS. The $E_2$ term of the ANSS is very hard to compute, it is a huge obstruction to progress. In fact, the Chromatic SS was developed in order to compute the $E_2$ term of the ANSS.

There are other methods though. The current approach seems to be studying the $K(n)$ and $E_n$ local homotopy groups of spheres by means of a Descent/Adams/Homotopy Fixed Point SS. These are what people are working on now, the main obstruction is, again, the $E_2$ term. For these we need to know the cohomology of certain Profinite groups with coefficients in some non-trivial modules.

3) I don't know about the Lambda algebra, but you should be able to write down a minimal resolution in a small range.

4) There is an approach to a large number off differentials in the Adams SS due to Bob Bruner. You take advantage of the highly multiplicative structure on the Adams filtration. This allows you to propagate differentials in low dimensions to get new ones. Its pretty cool.

If you are looking into Lambda algebra, the green book has some introduction to it. For some extensive examples you can checkout two papers: W.H. Lin's $\rm Ext^{4,\ast}_A(\Bbb Z/2,\Bbb Z/2)$ and $\rm Ext^{5,\ast}_A(\Bbb Z/2,\Bbb Z/2)$, and T.W. Chen's Determination of $\rm Ext^{5,*}_\scr A(\Bbb Z/2,\Bbb Z/2)$, where they worked out the $E_2$ terms of the Adams spectral sequence of spheres for filtration 4 and 5, completely with all relations. The results are rather complicated, but you can get the idea how it works.