Is there an issue with using loose, tea leaves on Shabbes (assuming we use kos shlishi, etc)-- does putting loose-leaf tea into a spoon like the one attached in the picture an issue of borer or considered cooking? Is there any other issue with using this on Shabbes?
According to Star-K article, no:
Since many people enjoy their tea mainly on Shabbos, it would be appropriate to briefly mention the do’s and don’ts of tea preparation on Shabbos. One should not use tea/herbal bags or loose teas on Shabbos. This is because tea is part of that group of foods known as kaley habishul, or easily cooked foods. These foods are considered so sensitive to heat since they will cook in circumstances that other foods will not.
Therefore, the only way to enjoy tea on Shabbos is to prepare tea essence before Shabbos. The common practice is to make a very strong cup of tea before Shabbos by using several bags and pouring boiling water over them. It is preferable to also remove the bags before Shabbos. This essence can now be added to a cup of hot water (kli sheini) on Shabbos. The preferred method, however, according to Rabbi Moshe Heinemann, shlita, Rabbinic Administrator of the Star-K, is to place several bags into a pot of water and actually boil them up on the fire before Shabbos. This essence, minus the bags, can be stored in a cup and added to hot water.
Alternatively, you may want to use instant tea on Shabbos. When using instant tea, or coffee, the preferred method is to add the tea to the hot water (kli sheini) and not vice versa.
See the footnotes in that article for sources.
I listed just one source. There are others that permit making tea using tea bags, and probably even using the loose tea in a cup.
However, I found no opinion that allowed use of the tea spoon as this is a special tool designated for straining, which is similar to the melacha of "borer".