Can I apply for a work permit or visa when I'm already in Russia with a 10-day tourist visa?
It is unlikely that you could accomplish that, as it takes longer than 10 days for a work permit to be issued after registration at the General Directorate of Migratory Affairs of the Ministry of Interior of the Russian Federation (GUVM, its acronym in Russian).
In addition, the employer requires permission to hire a foreign citizen. The best way to get a work permit is through the employer. Expatica has a useful guide on what is needed to move to Russia and how to get a work permit.
No. When a visitor applies for tourist visa, they declare tourist intention.
It is possible to get a work permit in 10 days, but when you apply for the permit, you should provide your passport. And if authorities would see a tourist visa you will be deported.
For work purposes there are special work visas. Any potential applicant should apply for them.