I'm looking for a way to have no sediment at the bottom of my bottles. Could I filter before it goes into carboy by placing a filter into a funnel, that is connected to my 3/8" tubing before it goes into the carboy?
And what sanitation precautions should I take? Just soak funnel and filter in Star-Sans?
No, that wouldn't work. The reason is that if you're bottling, the way the beer carbs up is through a mini-fermentation in the bottle. That means that the yeast will carb up the beer, then fall to the bottom.
In a well-made beer, though, that's a slight dusting and not an inch of sludge. If you wait and bottle the beer when it's clear (no filtering, just through gravity and/or cold crashing), the amount of sediment in the bottle is very light. Then, you let the beer sit in the fridge for a few days or longer, and the yeast compacts down pretty tightly. If you use a highly flocculant yeast, the yeast will fall out and form a little tight yeast cake on the bottom of the bottle.
If you absolutely positively want no sediment in the bottle, then the only way is kegging and bottling perfectly clear beer from the keg.
Some breweries bottle condition their beer, so if you've ever had a Sierra Nevada Pale Ale, that's the amount of sediment you should get with bottling properly.