Yes, you can use the same lancet to poke again, especially if it's the same poking session. Lancets do get dull, but 5 pokes should have been fine.
Lancets are very inexpensive, but my shelter had me soaking them in alcohol to reuse for several days. Forget that I said, and ordered a bunch online from ADW where the lancets are really inexpensive.
I used the same lancet for the whole day of poking. More than that, and they'd be too dull to use. Just don't share the same lancet between human and cat, or multiple cats if you have more than one cat you test. With reuse, it can take more force to poke and draw blood.
The insulin needle on the other hand is much finer gauge, longer, more prone to damage, bending, etc. The pictures that Red & Rover shows are for insulin syringes.
Yeah for testing in the sink! High up, easier for you to reach, small enclosed space to make kitty feel safe. Make it THE test spot. Hope there is counter space beside the sink to lay out all your testing supplies and treats for after. Bathtub is a little bit harder, trust me on that one (Wink liked to jump into the tub, especially on warm days).
Tell Smokey "Testing time" or something similar, and lead her to the bathroom. Before you know it, she'll be running there ahead of you. I take care of a friends cat, he needs pills, and he ALWAYS leads me to the bathroom for some attention and his flavored pill.