Those tea ball things suck eggs. a french press works fine but you'll have coffee carry over that effect the flavor. Most of the teas you have are good for western brewing the puer will benefit from multiple steepings but I wouldn't expect magic from those particular tou cha. For the teas you have I'd recommend a cup infuser like the type sold by Upton they will filter fine particles while allowing the leaves to move naturally and as the basket is removed after your steep time you may brew directly in your favorite cup and do multiple steepings with the puer easily. If you find you enjoy it and want to get into better oolongs, greens, and sheng a gaiwan can be had pretty inexpensively,(in the neighborhood of $10) and brew great tea.
Bearing the burden of responsibility..... It's probably my fault.
Treat your silver as if it were earthenware and your earthenware as if it were silver - Seneca, Letters of a Stoic