Wow. Okay, let me see if I can help with your questions.
1. Different branches of a family will often have similar but slightly modified coats of arms. Your example of Popham with a bezant between the buck's heads on the chief is a good example of that.
2. Seals generally link to individuals, and may be passed down through family generations. One thing to be careful of, though, is that I have seen a number of cases, especially in the 18th Century, where the seal used by someone on a document is actually a seal borrowed from someone else for that document. Sort of a "I don't have a seal for this document, may I borrow yours?" kind of thing. Just so that you are aware that this is something that happened.
3. The College of Arms may have more genealogical information than appears in other more public records, but the only way to know is to approach them and ask. Which, of course, may cost you something in fees.
4. Yes and no. In the examples you have of Berry, in each shield the Berry arms are those in the upper left square of the shield. The other coats of arms contained on the shields are those of heraldic heiresses who have married into the Berry family at one time or another. An heraldic heiress is a woman whose father has a coat of arms, but the father had only daughters (who do not normally inherit their father's coat of arms) and no sons (who do normally so inherit). In such cases, the sons of these daughters may quarter their father's arms (in the upper left and lower right corners of the shield) with their maternal grandfather's arms (in the upper right and lower left corners of the shield). This can happen more than once in a family's descent, and other arms may be added to the shield, replacing one or another of the quarters, and even increasing the number of quarters beyond four. A more detailed description of how this all works can be found at
https://en.wikipedia.org/wiki/Quartering_(heraldry)
5. There are a few armorials (listings of coats of arms arranged by surname) and ordinaries (listings of coats of arms arranged by the colors and charges on the shield) available. The most commonly used armorial is Burke's "General Armory of England, Scotland, Ireland, and Wales". Burke's has only entry for Bulkwort: Bulkwort (quartered by Berge, of Croscombe, Visitation of Devon, 1620). Sable a hand lying fessways within a bordure engrailed argent.
6. I am certain that others will answer with other possible sources, but you have already discovered that there aren't a lot of sources for genealogical research before the introduction of parish records. For tracing people in the upper classes (knights and above), there are the various peerages (Burke's Peerage, Baronage, and Knightage being the best-known but hardly the only one available). Searches through land records may sometimes yield fruit for you. If you are looking for a family with a coat of arms that goes way back, the four-volume set "Dictionary of British Arms: Medieval Ordinary" might help. Though it is organized as an ordinary, there is an index of surnames in the back of each volume. I checked, and didn't find Bulkwort, but Volume does list a name that may be close, "Bolkeworth, Sr William de," who bore the arms: Or a lion rampant purpure collared argent.
Anyway, I hope that all of this is of at least some assistance to you.
David