Why do these heraldic designs have weed leaves on them?
Well, why not?
There were a wide variety of "charges" (items categorized as objects that could be placed on the design, usually in a wide variety of places) used in coats of arms.
For many easily accessable examples, with one caveat I can suggest The Pictorial Dictionary of Heraldry at http://mistholme.com/pictorial-dictionary-of-heraldry/#alphalist . The caveat is that it is designed as a resource for armorial heraldry in the Society for Creative Anachronism, a not-for-profit for the study and re-creation of the European Middle Ages (pre-1601), which has certain ahistorical "in game" rules because reasons. But. The compiler of this work has put in effort to find early examples of the uses of such charges in the SCA period, usually near the start of the entry. Example:
An eel-fork, or eel-spear, is a tool used by fishermen in taking eels. It consists of a multi-tined head with broad, flat points; the shaft is not shown. Examples of the artifact can be found as early as the 16th Century; as a charge, the eel-fork is found in the arms of Stretele or Stratley, c.1413 [DBA2 350; cf. Parker 220]. The illustration is based on a period artifact.
Similar to the eel-fork is the "salmon-spear" or "harping iron", often shown in armory with a fish transfixed on its tines, as seen in the arms of Ränntl, mid-16th C. [NW 151]. It differs from the eel-fork in that its tines are narrow and barbed, rather like a trident head. (Nonetheless, this form was also, confusingly, blazoned as an "eel-spear" [cf. Guillim1 220]; there seems to have been no consistent distinction between the various forms.)
And that example is by no means the most obscure item ever seen on a coat of arms.
A lot of pre-modern heraldic treatises and modern heraldic merchants ("bucket shop heraldry") will talk about the Deep Symbolism And Meaning of each design element, but there's a striking lack of consistency about explanations, and a striking lack of testimony from the heralds of the heyday of heraldry. So basically I mostly call it a mug's game. To make something up on the spur of the moment, you say that a boar symbolizes persistence? OK, if it works for you, I won't contradict your opinion, whatever floats your boar. (The Aberdeen Bestiary at https://www.abdn.ac.uk/bestiary/ms24/f21v associated it with "untamed and savage".)
The patterns of Deep Symbolism And Meaning that I know of are:
One example is the turkeycock, at http://mistholme.com/dictionary/turkeycock/ :
It’s a period charge, found as the crest of William Strickland, 1550, who’s credited with introducing the turkey to England [Guide 189], and as the allusive crest of Robert Cooke, 1556 [Bedingfeld 101].)
So the first man's might be a merchant-like design, or more generally showing a relationship. The second is canting. (N.B. These are in crests, things sitting on tops of shields, not in coats of arms, what's inside the shields. Totally different. But charges used in shields were often used in crests and vice versa.)
Basically, searching for meaning in elements of a coat of arms is like looking at folk etymologies -- sometimes accurate and interesting bits of information, often a load of horse manure.
Background done, now to the query. There was such a wide variety of charges used on coats of arms, any new one representing an animal, plant, or object that was known to people of the time is entirely unsurprising.
At a quick glance at your examples: a quick search suggests that the very name Chennevières is derived from the French word for hemp, modernly "chanvre", and that hemp was grown in that area. I don't know that reliably. I only mention it as it may be an example both of canting and of merchant-type design for three of them in the top row. Others may be for hemp-growing regions, or because of some similar association, or maybe someone just liked the look. I don't know that the question is answerable in many cases.
Original Poster, is this sufficient or do you want to look in more detail on any of these coats of arms?