Can I apply Kirchhoff's current law to two separate nodes which are connected only by an ideal wire? 
Suppose there are 2 nodes A and B in the above circuit. If A has 2 incoming currents $I_1$, $I_2$, and B has 1 incoming currents $I_3$, $I_4$, is it correct to write the equation of whole nodes and circuit as $I_1+I_2-I_3+I_4=0$?
 A: For (truly) distinct nodes in a Kirchhoff circuit network, it is not appropriate to formulate the Kirchhoff current law taking in the currents from separate nodes and mixing an incomplete set from both.
For the specific circuit in your question, however, the "nodes A and B", as you have described them, are not actually distinct. Any two points linked by a zero-resistance wire are effectively the same node, so for your circuit there are effectively only two nodes. You should re-draw the diagram in a way that minimizes the number of nodes (and thus eliminates any superfluous nodes), and then you can formulate the Kirchhoff current law at each of the remaining nodes.
A: In your drawing A and B are not separate nodes, they are different points on the same node. All points connected by a single contiguous conductor are a single node.
If you tried to treat them as separate nodes then you would have to consider the conductor between them to be a 0 ohm resistor. This would do nothing but add some additional equations which would be more complicated but would eventually boil down to $V_A=V_B$, which we already knew by inspection of the circuit diagram.
