Is Kester 44 a multi-cored flux solder? Dave said in his video to get multi-cored solder.
Kester uses a single flux core, but it works extremely well regardless.

Anything from AIM (American Iron & Metals), Alpha Metals, Indium, MG Chemicals, and Multicore (Loctite/Henkel) are top notch performers as well. At which point the trick is to get the right alloy, flux type, amount of flux, and wire diameter that you need.

Just stay away from the cheap Asian crap on eBay, DX, and so on (genuine Goot is good, but very heavily counterfeited).
You may get more splatter from a single core vs. multiple cores due to larger voids/air bubbles (as it's not divided into multiple cores). The larger the air bubble/void, the bigger the splatter (and increased likelihood the operator ends up getting burnt). Fortunately, it's not as big a concern with smaller diameter wire IME (assuming a reasonable distance of your fingers from the iron's tip).
FWIW, I actually use a hemostat to hold the solder wire, as it help stabilize my hand for better feed control (reduces hand tremor). And it also happens to reduce getting burnt from splatter as well, as it keeps your fingers farther from the joint.

If I bought a 1LB roll for general purpose through hole and SMD work what diameter do you recommend? I was thinking Kester #66/44 63/37 .020" (0.5 mm)?
No smaller than .020", and no larger than .032" IMHO.
Kester offers .025" as well, and is what I tend to use the most. For larger stuff, you can make a wrap or two around your fingers and twist it into however a larger wire you need (i.e. good for tinning wire, soldering lugs, tabs, and so on).
Also, keeping a small bottle of RA flux on hand is highly useful as well (cheaper than disposable flux pens w/o sacrificing quality).
MG Chemicals Rosin 835 comes in small bottles (125ml & 1L), which is handy for hobbyists.

You can find vendors on eBay that repackage Kester flux (i.e. 186 RMA and 951 no-clean), but there's always a risk it's not genuine.
As per dispensing it, you can use needle bottles, brush bottles, dip a brush into a small container, refillable plastic nip pen, .... My preferred method is to use a refillable
BonKote BON-102 brush pen (eBay is full of counterfeits, so get it from a distributor if you want one; genuine ones start at $15 per and go up from there <multiple tip shapes available>).
FWIW, All-Spec carries Kester 44 as well so you can put all of it on the same order and save on shipping (good pricing & reasonable shipping fees IME; no affiliation, just a customer).
