Eh, no, a lot of people say you can feel it for example in shooters because everytime you make a mouse movement, it takes a while until you see it on the monitor, not even a second but in multiplayer this can be important for example. Must say I haven't tested it myself but I can see how it could potentially be annoying.
I'd say yes, don't know what mine has but usually anything below 8ms is fine, you cannot trust the numbers anyway as there are different methods of measurement and measuring grey-to-grey gives lower numbers than black to white or something like that, but if it's a TN panel there shouldn't be any tearing nowadays and all other panels seem to be above your budget anyway.
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