I can only speak of the fuzzy issue. I bought an unfinished IKEA table and chairs, it's definitely pine but I had fuzzies as well, I used the finest sand paper I could find and used some butcherblock oil to prep the wood before the stain since cheap wood tends to stain blotchy anyway. It helped to be able to rub off most of the fuzz with a microfiber cloth and I've had no issues since. I stained the legs but the top and chairs I used danish oil in natural finish.

Sandpaper causes "fuzzies". Using a wood scraper avoids that, however that's not a tool you can use without some experience. Use your sanding sealer and coat / sand / coat / sand until you get the pores of the wood sealed up as you like. That will knock down the fuzzies as well.

If you cannot use a card scraper, I'd recommend a coat of shellac, let it dry, then sand. The shellac will stiffen up the surface fibers.

What sandpaper grits are you using?

For clear finishes, I usually sand starting at 100 grit, then 150, then 220. After the 220 grit sanding, I wet down the wood--wet, not dripping/ puddling wet. Let dry & knock down with 320. Wet again, knock down, wet again, knock down.

With soft clean wood like you pictured, I would probably start with 150.

I've never used sanding sealer so can't comment there. Wood conditioner (which I use) is typically used on soft wood before staining so the stain will be more even. So far, I've used wood conditioner from the big box stores. It does help but isn't foolproof. Charles Neil's Blotch Control is supposed to be really good stuff but I have yet to use it.

http://charlesneilwoodworking.3dcartstores.com/Pre-Color-Conditioner--Pre-Mixed--1-Quart--Ready-to-Use_p_47.html?redirect=1