Not in the UK or Scandinavia it won't. The birch here is Silver Birch and in Scandinavia also Downy Birch. I have only encountered that smell with River Birch (read that its present in Sweet Birch too) which is native to the Northeast USA and Canada.If it is it will have a wintergreen/mint aroma.
I prefer to carve the wood when its green its a lot easier on the blades and the woods less prone to chipping. Now if you take care and place your piece in a carrier bag full of the chipping you have made and store it in a cool place it will slow down the drying process and the wood will usually be alright. Now I'm not saying it won't crack as the odd occasion it will but mostly it won't.Good effort mate, spent many a nights sitting out the back "making a mess" if you ask the missus. I made a kuska from a birch log like that but a word of warning. If it's not seasoned you may find it will split. I found out the hard way. Season 1st THEN carve.
I agree with Robson here on both points, also the op should get straight on to making their next one while this ones is drying.Back to the OP and the wonderful spoon #1 that he carved. Quick study. I think I can make a carver out of him yet.
1. The neck between the bowl and the handle should not be thinned. Looks good but that's a stress point with heavy mixing.
Stuff like bread, scones, pasta and pizza doughs. I stir with the handle. The bowl end would break my wrist.
The contours of the handle can be smoothed with the crooked knife, shallow, dragging, slicing cuts.
2. Thin the rim of the bowl void, from the inside. That makes the bowl look bigger and thinner. Yeah, it's a fake, an illusion.
Everybody should do it.