What fraction of air intake becomes bleed air in airliner engines?
Indeed about 1-2% of compressed air is bled off for cooling turbine disks and blade roots. When turbine inlet temperatures are higher than 1350K, the turbine blades must be internally cooled as well, and up to 15% of compressor air may be required as bleed air for cooling. Source



Bleed air is also used for aircraft systems. If we take a typical airliner such as the B737NG, compressed air can be supplied by either the engines, the APU or an external air chart, and is used for the following purposes:


Air conditioning &amp; cabin pressurisation, which are both subject to regulation quality standards.
Wing &amp; engine thermal anti-icing
Engine starting
Hydraulic reservoir pressurisation. The reservoirs are pressurised to ensure that hydraulic fuel reaches the hydraulic pumps. 
Water tank pressurisation


Compressed air is tapped from the engine 5th and/or 9th compressor stage. Air supply is regulated according to demand and varies according to flight stage - during flight stages with relatively high engine power (take-off, climb, most cruise), 5th stage compressed air only is used. During low engine power conditions such as approach and landing, 9th stage compressed air is used for the pneumatic systems as well. This air is under higher pressure, and hotter, than the 5th stage air.



Exactly how much bleed air is used for the systems is hard to get a reference for, but here is an estimate for the CFM56-7B18: 


Bypass ratio = 5.5:1, Air mass flow = 307 kg/s => hot flow = 307 / 5.5 = 55.8 kg/s
Cabin air systems refresh the whole cabin volume every 2-3 mins. Let's take 2 minutes, without recirculation.
Internal volume of a 737NG is about 400 $m^3$ (from fuselage width and length of an -800).
Air density = 0.94 kg/$m^3$ (we'll take cruise conditions, cabin air density @ 2,000 m)
Aircon flow = 400 * 0.94 / 120 = 3.1 kg/s


There are two engines, so the percentage of flow used for the aircon and pressurisation would be 3.1 / (2 * 55.8) = 2.8% maximum. Please again note that this is a ROM value which I will gladly give up for an official reference.

As a magnitude check: the APU brochure mentions 154 pounds of air per minute = 1.16 kg/s for environmental control and engine start. The APU can deliver air conditioning on the ground without the engines running so that would be the right order of magnitude.

Final estimate: bleed air for air conditioning is between 1 - 3 kg/s is about 1 - 3% per engine. 

Bleed air is not used for:


Hydraulic power generation. The hydraulic pumps are engine driven pumps (via a gearbox) and electrical pumps, powered by the AC system
Electrical power generation. This is provided by Integrated Drive Generators, from a generator in the APU, and from the batteries through an inverter.


Source: own work from open source on-line pilot training courses.