Who's dumb idea was to make Jar Jar
Fixed it for you.
But, yeah, no sense in him having a political career, but I bet they (Lucas) wanted to keep Jar Jar around in some capacity. But then again, Jar Jar is considered (likely) a Hero of the Battle of Naboo, and many politicians are former war heroes (not saying that they are all idiots since Jar Jar is).
I figure Boss Nass and the General (forget his name) may have died in the ten years between TPM and Clones since there's no mention of it... and maybe Jar Jar was influenced by Palpatine to become a senator since the Force influences the weak minded.
As for Padme: At the time she was hiding in Naboo, then ran off with Ani to rescue his mother on Tatooine, and then running to Geonosis to save Obi-Wan, leaving Jar Jar to be the senior member of the Naboo delegation.