P0977 - 2007 Toyota Corolla

The codes translate as.... (I'm assuming the first number given was a 'p' code as well).
p0010 = Camshaft Position Actuator Circuit (Bank 1).
p0700 = Transmission Control System TCS Malfunction.
p0974 = Shift Solenoid "A" Control Circuit High.
p0977 = Shift Solenoid "B" Control Circuit High.

The first code refers to the electronics controlling the Variable Valve Timing system in the engine. It has read a fault on the system that operates the valve settings in the engine.
All three of the other codes refer to issues with the electronics in the Transmission. Unless you have experience rebuilding transmissions, your safest option is to take the vehicle to a Good mechanic who knows transmissions. Messing with transmissions and the electronics for them without the correct knowledge can cause more VERY expensive problems.

DTC P0401 Exhaust Gas Recirculation (EGR) Flow Insufficient
Probably plugged passage's , not the EGR valve .
What type code reader are you using to read these codes ? I have factory service info. Doesn't show P0842 or P0642 , no such codes . You don't show what engine yours has , I'am assuming you have 3.5 L !
P0604
Control Module Random Access Memory (RAM)
PCM
DTC P0601, P0602, P0603, P0604, P0605, P0606, P0607, P1600, P1621, P1627, P1680, P1681, P1683, or P2610 in Engine Controls - 2.2L (L61)
or
DTC P0601, P0602, P0603, P0604, P0605, P0606, P0607, P1600, P1621, P1627, P1680, P1681, P1683, or P2610 in Engine Controls - 3.5L (LX9)

P0717
Input Speed Sensor Circuit Low Voltage
PCM
DTC P0717 in Automatic Transaxle - 4T40-E/4T45-E
P0741
Torque Converter Clutch (TCC) System - Stuck Off
PCM
DTC P0741 in Automatic Transaxle - 4T40-E/4T45-E
P0742
Torque Converter Clutch (TCC) System - Stuck On
PCM
DTC P0742 in Automatic Transmission - 4T40-E/4T45-E
P0748
Pressure Control (PC) Solenoid Control Circuit
PCM
DTC P0748 in Automatic Transmission - 4T40-E/4T45-E
P0751
1-2 Shift Solenoid (SS) Valve Performance - No First or Fourth Gear
PCM
DTC P0751 in Automatic Transmission - 4T40-E/4T45-E
P0752
1-2 Shift Solenoid (SS) Valve Performance - No Second or Third Gear
PCM
DTC P0752 in Automatic Transmission - 4T40-E/4T45-E
P0753
1-2 Shift Solenoid (SS) Control Circuit
PCM
DTC P0753 in Automatic Transmission - 4T40-E/4T45-E
P0756
2-3 Shift Solenoid (SS) Valve Performance - No First or Second Gear
PCM
DTC P0756 in Automatic Transmission - 4T40-E/4T45-E
P0757
2-3 Shift Solenoid (SS) Valve Performance - No Third or Fourth Gear
PCM
DTC P0757 in Automatic Transmission - 4T40-E/4T45-E
P0758
2-3 Shift Solenoid (SS) Control Circuit
PCM
DTC P0758 in Automatic Transmission - 4T40-E/4T45-E
P0815
Upshift Switch Circuit
PCM
DTC P0815 in Automatic Transmission - 4T40-E/4T45-E
P0816
Downshift Switch Circuit
PCM
DTC P0816 in Automatic Transmission - 4T40-E/4T45-E
P0826
Up and Down Shift Switch Circuit
PCM
DTC P0826 in Automatic Transmission - 4T40-E/4T45-E
P0973
1-2 Shift Solenoid (SS) Control Circuit Low Voltage
PCM
DTC P0973 in Automatic Transmission - 4T40-E/4T45-E
P0974
1-2 Shift Solenoid (SS) Control Circuit High Voltage
PCM
DTC P0974 in Automatic Transmission - 4T40-E/4T45-E
P0976
2-3 Shift Solenoid (SS) Control Circuit Low Voltage
PCM
DTC P0976 in Automatic Transmission - 4T40-E/4T45-E
P0977
2-3 Shift Solenoid (SS) Control Circuit High Voltage
PCM
DTC P0977 in Automatic Transmission - 4T40-E/4T45-E

Sometimes with code readers they get confused an just throw any old codes . Have it checked by someone with GM Tech 2 scan tool .