using led stripes with external battery: any side effects?
Yes, you can use you 12V battery to power your led stripes.

The Ikea stripes are internally connected that way :

12v
---------------------------------------------------------  -  -
            |              |               |
           ---            ---             ---
           | |            | |             | |
           |R|            |R|             |R|
           | |            | |             | |
           ---            ---             ---  
            |              |               |
            |              |               |
            |              |               |
           ---            ---             ---  
           \ /            \ /             \ /
           ---            ---             ---  
 GND        |              |               |
--------------------------------------------------------  -  -


This means that the same voltage ( 12V ) is applied to every resistor/led branches.
If you connect other led stripes the way Ikea recommends to. It's ok. You will still have 12V applied to every resistor/led branches.

The problem will be the current drawn by the system. If I assume that Ikea power adapter (0.25A) is design to power up to 4 stripes. Then, every stripe consumes :
$$ I_{stripe} = 0.25A/4 = 0.0625A = 62.5mA  $$

Thus your battery, which is able to deliver 2A can power :
$$ N_{stripe} = 2A/62.5mA = 32 units_{max} $$
32 units max. 

But what you really cares is for how long the battery will last. You have a 2000mAh battery. Thus, roughly, the time required to deplete the battery will be:

$$ t_{battery} = 2000mAh / (Nb_{stripe} \centerdot 62.5mA) $$

This is a rough approximation, because as the battery voltage drops, the drawn current drops also and the battery will last longer. But this is balanced a little bit by the fact that I have not taken into account that, after some time, the voltage will not be sufficient to light up the lights.

Another thing to notice : Here you don't have any voltage regulator if you use a battery alone. This means that the 12V at the battery output will decrease as the battery get depleted. And the emitted light will reduce over time.

Concerning the 4V lamps, you can connect 3 of them in series. If I combine this with the stripes:

12v
---------------------------------------------------------  -  -
            |              |               |           |
           ---            ---             ---         ---
           | |            | |             | |         |X| )4V
           |R|            |R|             |R|         ---
           | |            | |             | |          |
           ---            ---             ---         ---
            |              |               |          |X| )4V
            |              |               |          ---
            |              |               |           |
           ---            ---             ---         ---
           \ /            \ /             \ /         |X| )4V
           ---            ---             ---         ---
 GND        |              |               |           |
--------------------------------------------------------  -  -


You will have 4 V at each lamp.