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If I search on the same index which is currently being indexed/purged/optimized, will I get incorrect results?
optimized doesn't change the results; but purging will make all your entities unavailable until you don't index all your data again: it won't give results for the objects not indexed; so users may not find all results.
That's why I am telling you that you should rebuild your index in a separate index, and then replace them "hot" by implementing a DirectoryProvider.
(unless you are rebuilding your index at a time that the index being empty for a couple of minutes doesn't worry anybody)
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Or will it just make the search slow?
yes it will be a bit slower, just because when an index segment is changed some buffers need to be re-read from disk, but this happens all the time anyway and is usually not a problem.