All the lights flash but the engine will not start

SOURCE: 2000 vw bug shaking and check engine light flashing when driving

You have a really bad misfire going on. i would not drive the car but if you get it towed you can get the engine codes read which will give you more of an idea on whats going on. Otherwise its just a guess.

SOURCE: won't start after cleaning battery terminals

Sounds like you have an aftermarket alarm with starter disable. By disconnecting the battery the alarm reset and the defualt setting is usually to set the alarm automatically (active arming). You have three options..
A. If you have the key fob that went with the alarm you can disarm the alarm.
B. If no key fob is available you can disconnect the battery again, turn the ignition key to the on position and reconnect the battery. This usually doesn't allow the alarm to set and allows the car to start.
C. Remove the aftermarket alarm system. This is usually the long term fix i recommend if the fob isn't available. Option B should get you going though.

SOURCE: 2001 Toyota Avalon - Rear signals

Check the front lights. Usually if i flashes fast it indicate a bulb is out on that side.

SOURCE: Engine immobilizer system problem

you really need the dealer for this one. car is too new and should be under warrenty. This should not of happen. Most likely you have a computer issue with codes stored that will direct the technician in fixing this.

SOURCE: Left turn signals not working and left turn signal

one of the left turn signal bulbs is burned out. turn them on and look to see which one is out when you see one not on replace it and you are up and running again.