Can you use Effortless Lace & Lighten Weapon on double weapons and pole arms?
Double Weapons, no.
Benefit: Choose one type of weapon. When using a weapon of that type you may reduce the effort required to wield it by one step but suffer a −2 penalty to hit with that weapon. This allows you to wield a weapon 1 size category larger as if it were your own size, use a two-handed weapon in one hand, or a one-handed as a light weapons.
If I choose a double weapon I can reduce the effort required to wield it by one step.
(a) Is the double weapon 1 size category larger than the wielder? No.
(b) Is the double weapon two-handed? Optionally yes, practically yes, RAW no. RAW (see below) indicates a double weapon is considered two-weapon fighting rather than two-handed fighting.
(c) Is a double weapon one-handed? No.
To qualify for Lighten Weapon you'd need to be using the double weapon in two hands which (as the following states) is an option but also states it is not being used as a double weapon when wielded this way.
Double: You can use a double weapon to fight as if fighting with two weapons, but if you do, you incur all the normal attack penalties associated with fighting with two weapons, just as if you were using a one-handed weapon and a light weapon. You can choose to wield one end of a double weapon two-handed, but it cannot be used as a double weapon when wielded in this way—only one end of the weapon can be used in any given round.
So if you use the double weapon as it isn't intended and hold it in two hands and only use one end of it as threatening, then yes Lighten Weapon would apply. In this situation when Lighten Weapon applies you are now wielding a one-handed weapon and there doesn't seem to be anything which would prevent the Effortless Lace magical item from working in conjunction with that scenario. You might get into a semantics war over whether the text one-handed piercing or slashing melee weapon from the Effortless Lace description refers to a weapon wielded by a character in one hand after all modifications, feats, and magic has been applied OR a weapon from the weapons list under "one-handed" weapons, but that's up to the individual DM.
With polearms, assuming you mean these weapons (as outlined in Fighter's Advanced Weapon Training ability):
"Polearms: bardiche, bec de corbin, bill, Boarding gaff, crook, fauchard, glaive, glaive-guisarme, gnome ripsaw glaive, guisarme, halberd, hooked lance, lucerne hammer, mancatcher, monk’s spade, naginata, nodachi, ranseur, rhomphaia, tepoztopilli, and tiger fork."
Polearms mostly fall into the same situation as using the single end of a double weapon with two hands. Both Lighten Weapon and the Effortless Lace magical item should work together.
Will they work with the Two Weapon Fighting rules and Two Weapon Fighting Feats?
Too broad a question considering the number of feats involved which have to interact with one another.
Each combo probably deserves its own question. Covered Two Weapon Fighting and Weapon Finesse below.
Can I use them to hit with both ends of a pole arm or double weapon?
No. Effortless Lace requires a one-handed weapon and using a double weapon or a polearm as a double weapon (the other end just being the butt of the haft) is a two-weapon equivalent fighting style which uses both hands on the single weapon. It is simply not a one-handed weapon when being used as a double weapon.
Will two weapon fighting penalties be reduced to minimum if I use these in tandem?
Two-weapon fighting with what weapons? Double weapons? Oversize weapons? In one or both hands? To many options to consider for the vagueness of this question.
I basically want the effect of flurrying with pole arms/double weapons (a la Kilik and that new guy from Soul Calibur 1-6) without being a monk.
Flurry only grants one extra attack, as does wielding two weapons. Lessen penalties as feats and magical items allow.
Effortless Lace and Lighten Weapon don't work with double weapons.
They should both work with a polearm but that would not grant any extra attacks unless you wield an off-hand weapon as well.
Make the polearm with Effortless Lace on it your offhand (as with the Feat and magic lace it is considered light and one-handed) and you can wield another weapon for an extra attack in your primary hand but then you're two-weapon fighting with all those penalties which apply. [-4 to primary hand and -8 to off-hand]
Take Two-Weapon Fighting feat, Dexterity 15 requirement, and you're at -2/-2.
Use any one-handed weapon (in your primary hand) and the Lighten Weapon makes it a light weapon, and Take Weapon Finesse (Dex is at least 15 due to above feat requirement), and you're at +0/+0.
So, as I understand the feat and magic item, you could wield a longsword in your primary hand and a polearm (with Effortless Lace on it) in your offhand and by taking Two-Weapon Fighting and Weapon Finesse, you'd be at no penalties for both attacks.
Disclaimer: Any DM concerned with game balance should look sideways at the Player and ask them to justify how this situation can be explained by roleplaying and backstory.