Notice: Almost ALL of these algorithms below can be performed intuitively, and it is meant to. Do not purposely memorize them, but try to understand them. Just because an algorithm has an R2 doesn't mean you can't turn the cube around to see whats happening 
Edge Orientation:
Last edge orientation is affected by f2l, using moves from sub group <R, U, D, L, F2, B2> does not change the orientation of the last layer edges so you can use the 7 edge oriented algorithms.
You can search the wiki for algorithms solving the f2l. But don't be tricked because some of them using rotations such as y/y' which can actually cause edge orientation.
y' R = F
y R = B
Avoiding y2 rotations:
You might think that you have to y2, but you literally will never have to use it.
Sometimes you can use other types of rotations such as z, or even x if you know the algorithm.
I thought about what he said and tried to make an easier way to do these "harder" case:
Example:U F' U' F U' R U R'
Yeah, thats an easy case. Just do d (R' U' R) d' (R U R'). But what if its y2 away from you as shown below.
U F' U' F U' R U R' y2
Instead of y2 d (R' U' R) d' (R U R')
When can do these alternatives:
x L B D L' U' L U L D' L' (10f)
x' F U' F' U F' L F L' (8f*)
x' F2 U' F U F L F L' (8f*)
x' F U' R U' R' U F' U (8f*)
Instead of relying on y2 moves, you can use an x/z rotation.
Edit: ^ those algs are bad I suck with cube explorer.. but you get the idea.
EO Cross:
You might start objecting, saying this is a terrible idea, but EO Cross is very simple to do and plan if you have easy EOline cases.
The probability of having a 4 or less misoriented edges is a 27.44% Chance. You could possible take advantage of such an easy case and
perform EOCross.
Full EOLine takes an average of ~6.127 moves and a maximum of 9 moves
http://cube.crider.co.uk/zz.php?p=eoline
More use of 'EOCross'
Someone could use the idea from the Snyder Method, "The final CE pair is placed while simultaneously orienting and placing a minimum of one LLE. Then all LL edges are solved together with one corner. Then the last three corners are solved. "
ALL of the last layer edges would be oriented, so you would have less of 'his' cases!
But I'm not sure if Last Three Corner algorithms are finger friendly.
Thats a lot better than using my idea of Permutation ignoring OLLs for solving the cube.
F2L, Edge permutation preservation
You can use many insertion methods to preserve the permutation of Last Layer edges.
Hopefully you are using OCLL algorithms that preserve permutation.
This step can also be done intuitively.
Adjacent Edges(Back/Right):
U' R' F R F' U (6f*)
L' U2 R U R' U2 L U' (8f*)
Adjacent Edges(Back/Left):
R U' R' U (4f*)
L' U2 R U R' U2 L U' (8f*)
Opposite Edges(Left/Right):
U R U2 R' U (5f*)
All Edges:
U R U' R' F' L' U' L F (9f*)
R U L' U2 L U L' U R' L (10f*)
Lookahead vs Recognition
This is a widely debated topic(At least in this topic xD). While looking ahead in f2l can improve your times greatly, you cannot ignore the importance of improving f2l case recognition.
Recognizing the cases(as well as piece finding) faster allows you to require less look ahead, which means higher TPS. Its important to practice both of these f2l techniques to maximize speed.
Color Neutrality vs Fixed Cross
Typically, color neutrality has a few advantages to fixed cross solving(only using 1 color to solve the cross). While they both average +/- 1 the same amount of moves for a cross solve, you can have some easier
crosses to plan with color neutrality.
Single Cross:
*Allows the use of easier color recognition for cross, f2l, oll, and pll.
x:limited in its probability for a better start
Color Neutral:
*More X-Crosses, Shorter Cross Moves
*Easier Crosses
*Free F2L Pairs during inspection
*Offers better lookahead into F2L
x:Though widely trashed by white cross solvers saying it impedes recognition, IT DOES NOT. That would be like saying using green cross for 4x4 would slow down white cross for 3x3.
Edge Orientation:
Last edge orientation is affected by f2l, using moves from sub group <R, U, D, L, F2, B2> does not change the orientation of the last layer edges so you can use the 7 edge oriented algorithms.
You can search the wiki for algorithms solving the f2l. But don't be tricked because some of them using rotations such as y/y' which can actually cause edge orientation.
y' R = F
y R = B
Avoiding y2 rotations:
You might think that you have to y2, but you literally will never have to use it.
Sometimes you can use other types of rotations such as z, or even x if you know the algorithm.
I think I just convinced myself to alternate between bottom and left crosses during F2L. I've been messing around with this "alternating cross" strategy for the past hour or so and here are some examples of the many surprising things I discovered:
What do you do when you get this F2L case in the BL slot? You could do a y2 rotation then your usual algorithm. Or if the corner-edge pair happens to fall in the same slot then you could do L2' U L2 U L2' U2 L2 (kinda like a left hand Air Jeff). But what if it doesn't, or what if you're not comfortable with L moves? Do a z rotation then (R U R' U')*3 (the exact same algorithm!). And there's a bonus, if the corner-edge pair doesn't fall in the BL slot you could stop at z (R U R' U')*2 z' then AUF and insert it in the right slot.
Likewise for this case, if it was in the BL slot, you could do y2 then your alg. Or, z R2' U2 R2 U' R' U R' U2' z'.
z R2' U2 R2 U R U' R U2' z' in BL slot.
These are probably not new, but they were really shocking results for me, because I haven't seen anyone talk about this anywhere else. And I've always been doing the stupid y2 rotations, and now I'm suddenly relieved
By the way, should this be moved to the Speedcubing section?
Edit: if you want to experiment with my examples, here are the setup moves: First one is a pretty obvious self-inverse. Second one, R2 U R' U R U2 R2 U2. Third one, F2 U' F U' F' U2 F2 U2.
I thought about what he said and tried to make an easier way to do these "harder" case:
Example:U F' U' F U' R U R'
Yeah, thats an easy case. Just do d (R' U' R) d' (R U R'). But what if its y2 away from you as shown below.
U F' U' F U' R U R' y2
Instead of y2 d (R' U' R) d' (R U R')
When can do these alternatives:
x L B D L' U' L U L D' L' (10f)
x' F U' F' U F' L F L' (8f*)
x' F2 U' F U F L F L' (8f*)
x' F U' R U' R' U F' U (8f*)
Instead of relying on y2 moves, you can use an x/z rotation.
Edit: ^ those algs are bad I suck with cube explorer.. but you get the idea.
EO Cross:
You might start objecting, saying this is a terrible idea, but EO Cross is very simple to do and plan if you have easy EOline cases.
The probability of having a 4 or less misoriented edges is a 27.44% Chance. You could possible take advantage of such an easy case and
perform EOCross.
Full EOLine takes an average of ~6.127 moves and a maximum of 9 moves
http://cube.crider.co.uk/zz.php?p=eoline
More use of 'EOCross'
Someone could use the idea from the Snyder Method, "The final CE pair is placed while simultaneously orienting and placing a minimum of one LLE. Then all LL edges are solved together with one corner. Then the last three corners are solved. "
ALL of the last layer edges would be oriented, so you would have less of 'his' cases!
But I'm not sure if Last Three Corner algorithms are finger friendly.
Thats a lot better than using my idea of Permutation ignoring OLLs for solving the cube.
F2L, Edge permutation preservation
You can use many insertion methods to preserve the permutation of Last Layer edges.
Hopefully you are using OCLL algorithms that preserve permutation.
This step can also be done intuitively.
Adjacent Edges(Back/Right):
U' R' F R F' U (6f*)
L' U2 R U R' U2 L U' (8f*)
Adjacent Edges(Back/Left):
R U' R' U (4f*)
L' U2 R U R' U2 L U' (8f*)
Opposite Edges(Left/Right):
U R U2 R' U (5f*)
All Edges:
U R U' R' F' L' U' L F (9f*)
R U L' U2 L U L' U R' L (10f*)
Lookahead vs Recognition
This is a widely debated topic(At least in this topic xD). While looking ahead in f2l can improve your times greatly, you cannot ignore the importance of improving f2l case recognition.
Recognizing the cases(as well as piece finding) faster allows you to require less look ahead, which means higher TPS. Its important to practice both of these f2l techniques to maximize speed.
Color Neutrality vs Fixed Cross
Typically, color neutrality has a few advantages to fixed cross solving(only using 1 color to solve the cross). While they both average +/- 1 the same amount of moves for a cross solve, you can have some easier
crosses to plan with color neutrality.
Single Cross:
*Allows the use of easier color recognition for cross, f2l, oll, and pll.
x:limited in its probability for a better start
Color Neutral:
*More X-Crosses, Shorter Cross Moves
*Easier Crosses
*Free F2L Pairs during inspection
*Offers better lookahead into F2L
x:Though widely trashed by white cross solvers saying it impedes recognition, IT DOES NOT. That would be like saying using green cross for 4x4 would slow down white cross for 3x3.
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