Most lotteries are order-insensitive, and typically present the results in ascending numeric order. The actual draw often happens in some order (e.g. numbered balls being drawn from a hopper), and it'd be even more unusual if the numbers were actually drawn in consecutive order, but drawing 5-2-3-1-4 would typically be presented as 1-2-3-4-5 and would still be remarked upon as unusual.

Doing this, your final figures should be 12.1% and 96.8%.

For the expected number of times it would happen though, you are right, it would be three or four times per year on average for the second case.

Edit: What I did was take all the possible combinations of 5 balls (5!) by the number of different sets that could be drawn (50-5, based on lowest number), over the total possible draws (50!). I think perhaps what that does it not account for overlap between sets (1-5 and 2-6), inflating the number somewhat, which is why it's a bit more than twice the probability than you got for any possible sequence of 5.

  function rn() {
      return parseInt(Math.random() * 100);
  }


  function drawing(){
      let result = [rn(), rn(), rn(), rn(), rn()]; 
      return result.sort();
  };


  function sequential(arr){
      for(let i = 0; i < arr.length-1; i++){
          if(arr[i] + 1 != arr[i+1]){
              return false
          } 
      } 
      return true
  }

  let counter = 0; 

  for (let i = 0; i < 10000000; i ++){
      if (sequential(drawing())){
          counter ++;
      } 
  }; 

  console.log(counter);

  16