Transistor Amplifier Circuit (currents)
First of all is this schematic correct?


This circuit is a standard common-base amplifier using a PNP transistor.

If you wanted a D flip-flop, it's not correct.

If you wanted a PNP common-base amplifier, it is correct.


  What about current directions are they conventional? 


Unless you're in the US Navy or somebody specifically mentions "electron current", you can safely assume that "current" means "conventional current".


  I can't understand why collector current is heading to the transistor isn't it supposed to be output current


Your text is probably using the passive sign convention. This means taking all device currents going in to the device. Using this convention simplifies calculating the power consumption of each device in the circuit.

If current is actually flowing out of the collector, you'll just find \$i_c &lt; 0\$.


  why they are written in small letters when the are effected by the DC sources VEE and VCC


What small letters and upper-case letters mean is entirely a convention decided by each author. So you'll have to read your text to find out what is intended. 

It is common to designate AC currents with lower-case letters. It may be that even though DC currents are present, the author wanted to discuss AC currents, so he indicated the AC currents on the diagram.