It seems that you may be able to make your hot toddy using various stipulations. I'm addressing only the issues that you asked about. I can't think of others, offhand, so, obviously, address the scenario with your rav.
You mentioned the lemon slice, and this is addressed on Halachipedia:
According to many poskim, one should not place a sliced lemon in a cup
of tea because of the issue of cooking on Shabbat. According to
others, one may pour from a kli sheni onto lemon juice.
The sources cited for this are: Igros Moshe O.C. 4:74-18; Harav S.Z. Auerbach, quoted in Shemiras Shabbos K'hilchasah 1, note 150; Harav Y.S. Elyashiv, quoted in Me'or ha-Shabbos 1:221. And Halacha Yomit, citing Chacham Ovadia Yosef.
See p. 183 in this Google book (Shaarei Halachah: A Summary of Laws for Jewish Living, ed. Zeʼev Grinṿald). The conclusion appears to be that you may never measure exactly, which implies using a measuring tool. However, they state that even when using the measuring tool, if you don't fill it exactly, you are not "measuring". Of course, the best idea would be not to use any tool and approximate the amounts that you need by just pouring the ingredients. What could a little more schnapps hurt on Shabbat?
Not a problem. According to this article, en tzove'a b'ochlim - there is no concern of coloring when it comes to food. He continues to say that this is assuming that the coloring is to add flavor. You may not color the food simply for decoration, such as via food coloring.