How to get started with electronics and specifically cheap microcontrollers?
Firstly I would like to say that I'm quite happy to see someone so enthusiastic about electronics. 


Your choice of microcontroller. Although ATTINY45 seems good, reading the ideas of your pet projects I would recommend either ATmega168 or even better ATmega16L. These two microcontrollers are 4 times in cost as compared to ATTINY45 so it might cost 2 to 3 bucks in your currency.
What type of IDE are you going to use and what type of programming interface such as JTAG, ISP or Parallel Programming are you going to use?
With ATTINY45, you do not have enough pins to run LEDs although you might be able to debug with 3 atmost because it doesn't have enough pins
Although using 3 AA batteries would be a good idea for portability, they might run out fairly quickly and might cause unwanted problems when the batteries start to die out.


As for transistors, you can get plenty of general purpose transistors like the BC546 (my personal favourite) and YES you can run DC motors with transistors but small motors which you find in battery operated pencil sharpers, hand-held fans, etc as running motors will draw quite a bit of current which will make your transistor hot or even damage it.

If for instance you setup everything and things do not seem to work, you would need a multimeter. Its quite an essential tool when setting up the hardware. You do not need solder if you are using a breadboard. Breadboards can be quite expensive and you might want to go in for veroboard (or pre printed circuit board) for which you will require a soldering iron. I do not know what the prices are for tools like these in your country but if you do get a hot iron, do not save money as buying a good iron will save you money in the long run (speaking from past experience).

Let me know if I missed something