May we assume that you can "use" the logic elements more than once to create a digital "circuit" for your emulations?
Regardless, what you need is a four-stage binary ripple counter to count from 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000 which when applied to the binary coded inputs of the simulated "LCD" display will count 0, 1, 2, 3, 4, 5, 6, 7, 8. It will count higher after reaching 1000: 1001, 1010, 1011, 1100, 1101, 1110, 1111 which represents (in hexadecimal notation) numbers 9, A, B, C, D, E, F. Counts after F "roll over" to 0. Your "LCD" may not display counts greater than 9 correctly. The biggest problem I see is your available logic elements are combinatorial logic only. You need four flip-flops that you can "clock" to serve as a four-stage binary counter. Depending on how the simulator evaluates the states of the logic elements, you may be able to construct a D-type flip-flop using combinatorial logic and the "Memory Cell", but nothing is guaranteed.
If you really want to learn digital logic, I would give up this simulator and go play with real components. CMOS logic is fairly forgiving and readily available. I would start with a solderless breadboard, a few CMOS logic components, a handful of LEDs (so you see what the outputs and/or inputs are), and a handful of 1/4 watt film resistors in various resistances from 10 ohms to 10 Megohms. You might want to throw in a mixed bag of electrolytic, ceramic, and film capacitors of various values, and a few signal and power diodes. All this "stuff" is available in form of electronics tutorial kits, most of which you can purchase online for considerably less than a hundred dollars or a few dozen quid. If you do decide to putter around with digital logic, purchase a CMOS logic probe. This makes it easier to poke around to find the binary state of inputs and outputs. A really good logic probe will also have a pulse capability that can force a logic zero state or a logic one state for a few nanoseconds, which is really handy for testing counting circuits, flip-flops, etc.
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