When I was there mid-July in 2005 the blue route from Stary Smokovec was still closed because of last year's storm. You'll have to take the green to Hrebienok then the red to the point where blue is accessible. This adds only 15 minutes each way to the climb.
does not cover all the Tatras trails.
For Slavkovsky stit there is one official exception: in time of not being covered by snow it is legal to summit Slavkovsky by trail from Slavskovska vyhliadka (the same rule is applied to ascending Ostrva - Osterwa from both sides - from Popradske pleso and from Sliezsky dom). One can see - it is a very vague formulation - to differ properly, when it is uncovered by snow and when it is covered by snow. It is usually understood, that in times of trail not being completely covered by snow, you are allowed to go. But it is also popular summit for winter (ski)alpinist ascenders. It's mess in rules and practical obedience to them, but it is useful to know about this legal exception. Compare with this official link (only in Slovak): http://www.tanap.org/download/navstevny-poriadok.doc
BTW I heard some park official talking about allowing mountaineers to ascend some mountains also in winter:) Let's hope this to come true, but let's be neat to the nature we admire!