hi Aperson
now home from work and with a decent internet connection, I was finally able to see the videos. As with your diagram in your second post therer's definately no capacitor bypassing done of the PSU, as Duke37 commented about in your drawing.
This is really essential. if you are using a 7805 type voltage regulator as in the videos, you should have a 100uF in the input to the regulator and a minimum of 7V there. Preferably 8V to 12V to ensure that the regulator produces a stable 5V output.
On the output you should have a 10uF electrolytic or tantalum capacitor.
Both the 100uF and the 10uF should be mounted as close as possible to the 7805 reg.
For good practice it is normal to have a 0.1uF or 0.01uF right at the Vcc pin of the chip.
All those cap values and locations mentioned are the good standard practice for digital circuits. have a good look through old computer boards and you will find 0.1 or 0.01 decoupling caps on the Vcc rails of virtually every chip.
Now fiollow Duke's other comments re the connections between the LM35 and the ADC0804 and see how you get on

report back with hopefully good news
cheers
Dave