Minecraft – Why is it snowing in a forest

What you should do is make the water only one block deep, so that it all levels out. See the question that ZoogieZork posted. After that, you can dig as deep as you want and the currents shouldn't appear.

In general, here is a good technique for building rivers (top-down view):

█▒█▒█▒█▒█▒█▒█▒█... | █ = block
.█▒█▒█▒█▒█▒█▒█▒█.. | ▒ = water source
..█▒█▒█▒█▒█▒█▒█▒█. | . = empty block

First, you want to dig out a path for your river. Place alternating blocks and water sources along where your river is dug out so it looks like a checkerboard. Then, destroy all the blocks. Because there are water sources every other block, the space created by the blocks being destroyed will turn into source blocks themselves and create a nice and smooth river.

In your case, you would probably be able to build this pattern right up to one block before the lake. When you've destroyed all the blocks and the river is nice and smooth, you should be able to connect the lake and the river and the connection between them should level out.

Your best bet is to place torches or glowstone (or even fire or lava) a sufficient distance in the air (using lampposts) that the light level at the ground isn't sufficient to melt the snow, but is sufficient to stop mobs spawning. The answers in this question should help you with this calculation.