Eliminating D3 and LA1, the remaining circuit is a full wave rectifier using a center tap transformer. D1 and D2 are diode but you have put light emitting diodes (LED's). If LA1 is the lamp then putting D3 in series with lamp will prevent the lamp lightning up and LED will restrict the current. You will be seeing only LED glowing not the Lamp and in your case voltage is 20 so LED will burn out.
This circuit can not be made as digital, why because all digital circuits have this circuit in the beginning plus a capacitor in parallel to make it DC. The digital circuit use this DC voltage to RUN. How can a circuit Digitize its own power supply? I mean before it can get started. So every digital circuit has this analog part.
Some circuit can not be made digital just like the one you discussed. Digital circuit allows us to save our information and to process it easily. Example if you have signal of random shape starting from 0sec and ending at 1 sec. if you take one point on that time interval say 0.5sec, you can not determine the shape of the signal. so if you take point at every 1ms of 1sec time interval, you might understand its shape more clearly. So as you keep on increasing those points/samples you will have a better shape. The gap between those points/samples is your loss of information as you have no data between those points or samples. So some circuits can not bear those loss of information or it makes their efficiency less. If you get infinite points/samples, you have a analog signal. Do you thing that there is such a device that can store infinite samples? Even if you see digital oscilloscopes they are rated as 250MS/s and more than 2GS/s. means they can take 2 Giga Sample per second.
Your temperature sensor is analog but you use ADC which converts the signal to digital.How many points,well it depends on the bit resolution of ADC, ADC0804 is 8bit so you get 2^8 -1 =255 points.