LED current at output of AND gate
Your analysis of the problem is correct. You have chosen a poor AND gate circuit - but one that is promoted in many tutorial sites.

A couple of comments on your schematic:


The battery is connected backwards.
The convention is to have the positive rail at the top, ground at the bottom.
Circuit should generally read from left to right with current flowing generally from top to bottom.


Your circuit would be close to correct if you rotate it 90&deg; clockwise.



Figure 1. An AND gate made of a NAND gate followed by an inverter. Source: Circuits Today.

Why is this better?


Switching on X or Y on their own will not allow current to pass from R3 to ground. Q3 will be biased on and Z will be 0 V.
When both X and Y are on the collector of Q1 will be pulled low. This is a NAND operation.
When Q1's collector is pulled low Q3 will turn off and Z will be pulled high by R4. Q3 is forming a NOT gate (an inverter) and a NAND followed by a NOT is an AND.


You will see a similar arrangement in some logic series.



Figure 2. A CMOS AND gate. Source: All About Circuits.

Notice that the AND gate is more complex than a NAND gate and there will be some propagation delay due to the inverting stage. The linked article should be worth a read.



A few comments on your updated schematic: 

You've got the top to bottom right now but your schematic is reading right to left. Here's a flipped version (but I didn't flip the text). Some of the points you had addressed already but I'll comment on them anyway.



Figure 3. Rough schematic edits.

As explained, convention is that current flows from top to bottom so positive rail on top. For reading left to right we generally put the inputs on the left and the outputs on the right.


Battery right way up.
+V on top.
Since you've got grounds on R4 and R5 you need to indicate what these are connected to. Here we add the GND symbol to the negative rail. All the grounds are now connected.
We'll draw the LED and resistor vertically. Now it's clear that current will flow through it from top to bottom. You don't have to be too strict about this but in this case the clarity gained is worth the little bit of extra height on the schematic. In this case LED3 is the output of the circuit so it makes sense to locate it on the far right of the circuit.
Transistors right way up. For these NPN transistors the emitter arrow shows the current direction and so we have them pointing down towards GND.
Same as 5.
Input switch on the left.
Vertical LED is good. Vertical LED would be nice too but would add to height. Current flow is quite obvious in this case.
The GND symbol represents a cut section of the ground rail which is normally drawn horizontal. For this reason the symbol should be inserted as an upside-down T. The other 'earth' symbol (see Ground, earth and chassis explained
 written by me) represents parallel plates buried in the ground so there is a correct orientation for it too.