How pure NaCl is typical table salt? How is it purified?
Table salt is often iodized (iodine is an essential micronutrient, and this is one way to introduce it into the diet in the right amount). According to Wikipedia,


  Edible salt can be iodised by spraying it with a potassium iodate or potassium iodide solution. 60 ml of potassium iodate, costing about US $1.15 (in 2006), is required to iodise a ton of salt. Dextrose is added as a stabilizer to prevent potassium iodide from oxidizing and evaporating. Anti-caking agents such as calcium silicate are commonly added to table salt to prevent clumping.


Elsewhere, Wikipedia mentions other anti-caking agents, like sodium ferrocyanide, sodium aluminosilicate, and magnesium carbonate. And they point out that salt may also be "fortified" with iron or other micronutrients.

The certificate of analysis for "ultra pure NaCl" from American Bio shows TRACE amounts of aluminum, arsenic, bromides, heavy metals, iron, magnesium, phosphates, and sulfates. Those impurities may well be expected in table salt, too. 

From all that, you'd think that table salt is fairly impure. Let's look at the numbers to put the amounts of these impurities in perspective. The box of Morton iodized salt in my kitchen gives me the amount of sodium per serving. If all of the sodium is in the sodium chloride, I estimate the purity as

$$\frac{0.590~\mathrm{g}~\ce{Na+}}{1.5~\mathrm{g}~\text{table salt}}\frac{1~\mathrm{mol}~  \ce{Na+}}{22.989769280~\mathrm{g}~\ce{Na+}}
\frac{1~\mathrm{mol}~\ce{NaCl}}{1~\mathrm{mol}~\ce{Na+}}\frac{58.44277~\mathrm{g}~\ce{NaCl}}{1~\mathrm{mol}~\ce{NaCl}}\cdot 100\% = \fbox{99.99% NaCl}$$

where I'm assuming the label information is actually a bit more precise than reported (exactly 1.5 g of table salt contains approximately 590 mg Na). This is just a BOE (Back of the Envelope) calculation; I should round my answer to three or possibly two significant figures. 

In fact there are sometimes other sodium salts present (anti-caking estimates such as Tivolex contain sodium) so this purity is a slight overestimate. One salt company (WA Salt supply) lists "typical" purities for the table salt it provides at 99.72%; this salt contains uses Tivolex. Morton Salt claims that their iodized table salt contains 0.04% dextrose and "less than 0.5% calcium silicate", so the mg Na per serving on the label is a little bit higher than it actually is; I should get something above 99.46%, considering that additional information.

Still, that's pretty pure. The assay for "ultra pure" NaCl linked above gives 99% to 100.5% NaCl. Sigma-Aldrich has reagent-grade NaCl (>98%), ACS grade NaCl (>99.5%) and TraceSelect NaCl for doing heavy metal analysis that is more than 99.999% pure.

Is 99% purity "pure enough for high level experiments in chemistry?" There is no one answer. It depends on what those experiments are. How many significant digits do you need in your results? Are the results going to be affected by the presence of tiny amounts of iodide, iodate, glucose, calcium, or silicate, or the other impurities listed above? Maybe, maybe not. If a specific impurity interferes, find a way to lock up, remove or otherwise neutralize its effects.