A FOE! Common and glossy buckthorn are invading North American woods and landscapes, and destroying existing systems. Cut it as soon as you can. Google "invasive plants" and you'll find buckthorn on most state and provincial lists. Here is a link that might be useful: images

floraluk- 'Flowers may have a scent or not. Leaves may have a scent or not. Flowers and leaves on the same plant do not necessarily smell alike.' So, I guess I can conclude it is not the Cherry Laurel because the leaves should have a scent regardless of whether it flowers?

@hosenemesis - Thanks for the great ideas. BTW the one landscaper told he meant "Podocarpus" as he said Privet Wax Leaf and Ligustrum is the same (?). The reason, however, that he is pushing the Texas Privet is that he said it's the hardiest of the 3. I didn't think of the fact that the ivy would grow into the shrubs and yes, it would make a mess. The barrier makes perfect sense but I imagine it won't be cheap to dig up that ivy, cut it back, and then make sure the roots are really gone otherwise if it starts coming back once the hedges are planted, ugh. I'd get rid of the ivy but I think we have it and neighbors too as it helps with the erosion of the sloped portion of the backyard edge. @yukkuri - My wife would love citrus and avacado trees but do they really grow together enough to provide privacy? And won't they get wide enough to take up a ton of room along that wall (the wall is probably about 100 feet or more)? I suppose those types of trees would at least live better with the ivy. @desertdance - Thanks for the Sour Orange suggestion. I'll take a look at it and ask someone at a local nursery. And I thought this part of home ownership would be the stress-free part. Now I'm almost frozen trying to figure out what to do as it seems there is strong potential for regret or a do-over due to so much I simply don't know...and you hope the landscaper suggest what is sustainable and not just to make $$$. Especially the fact that neither of the two quotes said anything of keeping the ivy away from mingling with the shrubs which makes sense that it would happen...and be a headache.

Melaleucas have been problematic in parts of Florida because they have become invasive, spreading particularly well into the vast Florida Everglades. The wet summers and dry winters have allowed them to take over. A similar problem occurs with many of the species in parts of South Africa, where they have invaded large areas and continue to spread. Here in California with our Mediterranean rainfall pattern and dry summers, they have not tended to spread into the wild or become problems, even in summer irrigated gardens. We certainly don't grow the vast amount of species that exist in Australia, but they are very popular trees and shrubs in much of California, and are almost ubiquitous in shopping malls of Southern California, where M. quinquenervia in particular is much used. Several species of Melaleuca were also used to help stabilize sand dunes when San Francisco's Golden Gate Park was first built at the turn of the last century. They are generally considered relatively trouble free plants here in California, but most aren't hardy enough to be used away from the coast. Among their attractive qualitites here are nearly year round or winter bloom, and their ability to survive and thrive on zero supplemental irrigation once established. However, several species are extremely intolerant of drying out as container plants, and require careful nurturing once planted until they root in. I have seen species such as M. linariifolia quickly die if not watered daily as container plants, and they do not revive if allowed to dry out. Same thing with Leptospermum scoparium here. We tend to think of them as all being very drought tolerant under our conditions, so it is rather surprising to know that many grow in seasonally inundated areas in habitat.