Of course you can, depending on the solvent you will dissolve more components. However, if your goal is to extract the essential oil, you should choose another method. I don't think that hydrodistillation is a good idea, because the essential oil can be sensible to this temperature. I think that the extraction using supercritical fluids (CO2) is the best option, specially if you plan to use the extract for medical tests or similar.
Good option, You can use it, You have to be careful while selecting solvent, if you have expected compound, it will be also good. then use solvent whose BP of expected within range.
Using Soxhlet apparatus may be one of the ways to extract the compounds from Lemongrass, but it depends on what type of compounds you are targeting. Personally, I prefer using cold solvent extraction (maceration) over the Soxhlet; because, in Soxhlet, the extract is continuously exposed to heat, which is not recommended for natural products that are thermolabile. If you are searching for new compounds, maceration employing organic solvents is one of the best techniques. However, if you have the facility, try for supercritical CO2 extraction, which yields better aroma extract. Further, the main compounds of essential oil produced by hydrodistillation of lemongrass (Cymbopogon flexuosus) are genaial, neral, myrcene etc., which produce lower oil yields compared to solvent extraction.
It is not the question you should use the soxhlet or not. Find out what are you looking for in the extract. I would suggest that you set up one set of experiment with soxhlet and another set by simply adding the respective solvent and may be putting it on a shaker at room temperature. Now (may be after suitably concentrating) run TLC and see the difference in the number of spots and then decide which of the methods is more suitable for your investigation.