Few places in Denmark are more safe than North Zealand, indeed few places in the world are. In the forests however you may be bitten by ticks, and a fairly recent development is that some ticks now carry the more dangerous east european TBE until the mid 2000's Bornholm was the only area in Denmark with ticks carrying this disease. That said the risk of being bitten by a tick is minimal, and the risk of the tick carrying something serious is minimal too.
Danish healthcare is modern and efficient, although in some cases very bureaucratic and expensive. Usually you can simply rely on this, but if you consider serious hiking or trekking in forests of eastern Denmark consider vaccination against east european TBE.
talk
Talking is never likely to be a problem, neither will actually being understood by the locals in this part of the country. Nordsjælland is by and large populated by a affluent, highly educated, well travelled elite - you might even get by in your snobbiest French up here.