1) Would extension tubes be of any benefit here, since the camera/lens cannot be moved closer to the object? Stated in a general way, if the distance between the lens and object remains fixed and an extension tube is added, would the magnification increase? I am aware of the rule that magnification increases by the length of the tube divided by lens focal length, but does this assume that the lens is moved closer to the object?

No, extensions tubes would not help at all. Extension tubes do not change the optical characteristics of the lens, they merely move the camera further away from the lens. This lets the lens focus closer than it could before, which has the corollary effect of increasing magnification. So the extension tubes' magnification works at the expense of subject distance — the exact opposite of what you're trying to achieve.

2) If extension tubes will not help, are teleconverters the only other option? I have used a 3x teleconverter in the past with the above mentioned 210 mm zoom lens.

Theoretically, teleconverters could be an option, but you have to understand that teleconverters also do not change the optical characteristics of the lens they are attached to; they merely magnify the center of the image circle that is projected onto them. So without moving the lens at all, if you added three (3) 2× telconverters to your lens, you could theoretically achieve an 6 * 0.2 = 1.2:1 magnification, at the lens's minimum focus distance of 90mm. If you focus the lens a bit further away (exactly how far away, I can't say), you can bring down the magnification to 1:1, and buy yourself some more working distance. If you had more teleconverter power, that would buy you more magnification, which would result in more leeway to move the system further away from the subject.

However, stacking 3 or more teleconverters will result in pretty substantial image quality loss. Not to mention the light loss of 2 stops of light per 2× teleconverter = 6 stops. Assuming you don't have the luxury of increasing the shutter duration by a factor of 64, and also assuming you're working at or near the widest aperture your lens will allow, the only ways you make up for that loss is to crank the ISO by 6 stops (say, from 100 to 6400), or add a whole lot more illumination to the subject being photographed. But because you said you're capturing laser-illuminated droplets, I assume you can't increase the laser power by a factor of 64.

So really, the best option is simply use a longer focal length macro lens. While it's not cheap, the AF Micro-Nikkor 200mm ƒ/4D IF-ED can be rented for around $100 for 7 days at places such as LensRentals.com. It has a 1:1 reproduction ratio a MFD of 1.6 ft (488mm). You will be much happier with the results than trying to "lego" together a bunch of teleconverters to the back of a smaller lens.

Edit: Even with the AF Micro-Nikkor 200mm lens, you probably won't be able to achieve 1:1 magnification at 288mm. The "minimum focus distance" (MFD) of lenses are the subject-image distance (i.e., from the thing you're capturing, to the image sensor plane). The working distance is the MFD, less the lens's length, less the additional distance between the lens to the sensor (roughly, the "camera body thickness"—this is not a precise description of that distance).

Unfortunately, lenses' overall lengths tends to include the bayonet mount and additional electrical bits sticking out, so it's not as simple as just accounting for flange focus distance (FFD) for a particular lens mount. But for Nikon F-mount systems, the electrical contacts stick into the body about 6 mm, so the the distance between lens and image sensor is about the F-mount's FFD (46.5 mm) less 6, so about 40.5 mm. Thus, the actual 1:1 working distance of the AF Micro-Nikkor 200mm is 488 – 40.5 – 193 (the lens's length) = 254.5 mm

In order to make up for the missing ~34 mm, a low-power teleconverter (or low-diopter close-up lens, such as StephenG talks about in his answer) (such as the Canon 500D, which is just a +2 diopter), would be more than sufficient to give you 1:1 magnification, and also increase your working distance.