How good are my chances for snow at that time?
It is very likely to encounter snow in Slovenian mountains in March. You may still find patches of snow in June, perhaps even later. In fact, you should be an experienced hiker and have full winter hiking equipment if you plan to visit. You should also be fully prepared for the hike, monitor weather reports, ensure mountain cabins are open, etc. Please, do not take our mountains lightly. Lives are lost up there, even experienced hikers and mountain rescue team members may succumb to the dangers encountered there, especially in winter conditions and the start and end of that period (varies by location).
This is a translation of weather forecast for Julian Alps for March 20th, 2016:
It will be cloudy and often foggy tomorrow. Expect occasional light snowfall. South-eastern wind at first, which will turn to south-western wind during the day. Temperature at 1500m will be 1°C, -7°C at 2500m.
Weather Forecast for Julian Alps for Tuesday, 20th of March, Slovenian Print Agency, Ljubljana, 19th of March.
Did you know that up to 15th of March all vehicles on Slovenian roads (entire country, except the coastal region) are required by law to be equipped with winter equipment (winter tires or chains)? That is because we may get snow in March even in lower regions. I remember freak snow storms with 30cm of snow in May from my childhood - and that was at ~300m of altitude (the norm here).
Slovenians are a nation of hikers, jokingly, you are not considered to be a "true" Slovenian, until you have visited our highest peak - Triglav. You can preview hiking trails on specialized websites, such as hribi.net (most of it is in Slovenian, but includes images of entire trails) and Alpine Association of Slovenia where you can also get reports on conditions and much more (including where to find educational classes).