Can I test for lead with household chemicals?
General remark: get a bit of lead (metal or salt - whatever you look for) to have a positive control for your experiments.

More non-chemical tests for lead (metal): 


lead is soft: you can cut/scrape it quite easily with a normal knife. 
sheets of lead are easily bent, and they stay so. 
For bigger pieces: Hammering on it would leave a rather big impression of the hammer. The sound would be dull.
it usually looks grayish, not silvery-metallic, but the cut will be silvery.


If it is about the paint:


what colour is it? 
white lead: $\ce{2 PbCO3 \cdot Pb(OH)2}$ I'm more familiar with that in the context of paintings or painting of wooden things (window frames)
$\ce{PbSO4}$ also white. I think it has also been used for things like window frames.
side thought: lead paints for wood may have been popular due to the fact that the toxicity to microorganisms helps very much with the anti-fouling effect. Anti-corrosion paints for metal need different properties.
lead yellow: $\ce{PbCrO4}$ Wikipedia says that was the original pigment for school buses. 
$\ce{2PbO \cdot PbO2}$(Mennige) red anti-corrosion pigment that was at least here in Germany the most important lead based priming paint. Only difficulty with recognizing by color: lots of non-toxic replacements are produced with similar colour to have the familiar look...
Very sensitive but but not very distinctive test$^*$ for $\ce{Pb^2+}$: the sulfide is black and very insoluble. Kitchen chemistry: put some egg on and even the surface of the metal should turn black. Possibly some heat is needed (I didn't test, though). Otherwise, use acidic $\ce{Na2S}$ solution from the school lab. If the $\ce{Na2S}$ solution doesn't give a black precipitate, there could in theory still be lead in the paint which is not accessible by the sulfide due to the binding material of the paint. In that case, however, the toxicity will be less of a problem, too (i.e. unless you dissolve the paint somehow, burn it, ...)
$^*$ OTOH it is highly improbable that there are substantial amounts of silver in the paint...
reductive borax bead should be gray, oxidative colourless.
Isolating Pb is quite easy, maybe you can do that in your school lab: precipitate $\ce{PbCl2}$ with cold HCl, filter. The dissolve the chloride in hot water. You can then precipitate PbS (black) or $\ce{PbCrO4}$ (yellow).  Looking at the crystals under the microscope can help further. Qualitative inorganic textbooks often have microscope photos of different precipitates.