Re. the original q - there's nothing against greeny blue but it would be expensive. Surprisingly few ways to get a green from one plant source - apart from dyer's greenweed and one or two others - and the way it was got, historically was usually over-dyeing woad with weld or vice versa. By doing this you're doubling the amount of work for yourself so the price of such cloth would reflect the extra work gone in to getting that colour... So the short answer is - to a good dyer, very little would be 'impossible' but many things 'less feasible'. No reason why you couldn't get a bluey green at any period in history but you may have to have the status to afford it?
Re. kermes, Jill Goodwin says:
Kermes is made from the small dried bodies of the female shield louse (coccus ilicis), which lives on the leaves of the kermes oak...and the holm oak...found along the coasts of the Mediterranean Sea... Latin scholars referred to it as Grana tinctorium, for ancient dyers believed that the insect emerged from a grain or berry and referred to wool dyed with kermes as 'dyed in the grain'....
Apparently you use it as cochineal. I'd love to know if any other natural dyers here have ever had the chance to try it, and what results you got?
I have read elsewhere it was used as a cheaper alternative to Tyrian purple - that it's more a fuschia-mauve than a true purple or true red but I have no idea, never having seen it.
Sounds like Lord Howard's crimson was kermes, then or based on it - it's possible other things were added to 'stretch' it a bit as it wouldn;t be cheap. You're too early (I think?) for tin mordant which by the 17thC was added to madder dye baths to shift the reds to a more vivid shade... but I imagine the red you'd get from kermes would be quite a nice one, without that? There's no way kermes would be a cheap material at any period in history so I'd imagine it's poss they stretched it with some madder. They were always looking to improve reds. Once they mastered using different mordants with madder, that made things like kermes even less relevant I guess? Either way, he's not being mean, there - presumably the liveries reflected his status, so the posher the better?
How you dye with it - is pretty straightforward - just a boil up. No fermentation like, say woad, and you wouldn't have to be massively skilled. Although madder is much cheaper as always was as it was easy to grow... it's actually more skilled to dye with, as you need to pay attention to the alkalinity of the dyebath AND keep your temperatures lower than boiling point - or you lose the clarity of the red. As I say it's possible they stretched the kermes with madder, as they often did that with dearer dyestuffs.