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Put aluminum foil down to avoid getting rust on your food. Also, scrub really well with a grill brush!
Not if you can't clean the rust off
A wire brush and steel wool and/or a scrunge-type pad along with plenty of hot soapy water should remove the rust. If not, take the grate to a mechanic's shop and ask them to remove the rust. They'll use a wire wheel and plenty of pressure. Once cleaned off, wipe the grate down with cooking oil.
You may need to re-season the grate. To do that, simply light your grill, get the grate good and hot, then with a cloth or wad of paper towel dampened with cooking oil and held in tongs, wipe the hot grill a few times. You'll get lots of smoke and maybe even some flame, so be careful.
To help keep your grate seasoned, heat the grates as above, scrub off any residual food from the last cooking session and oil the grates again as above.
Iron, in the form of iron oxide (rust), is not readily absorbed by the body and is therefore of no concern when ingested from cooking surfaces. Conversely, rust removers tend to be quite toxic so I'd avoid using them on a grill. It sounds to me like you're not using your grill enough. Follow WileyP's suggestions and get cooking!
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