Preliminary research has answered a few of the questions listed above.

1. The grant of these two Arms is described in the preceding entry in the General Armory.  They were awarded through the Office of the Ulster King of Arms, which had the sole power to award and renew Arms throughout England, Scotland, Ireland and Wales.
2. Most Arms included helmets, and the type and orientation of the helmet was used to indicate the status of the individual or family.  A steel helmet with closed grate indicates a status below even a Knight ... that of a Squire or a land holder.
3. The insignia of a cross indicates some connection with a religious organization, while the lion might indicate service related to the King.  There were occasions in Wickenden history, now described in the section on Knight Service, where a Wickenden provided Knight Service to the Prior of Michelham as well as to a King and Queen.
4. The Augustinian Priory of the Holy Trinity was founded at Michelham in 1229 by Gilbert de Aquila.
5. The Wickendens were owners of the Manor of Lewisham as copyholders during the reign of King Philip and Queen Mary and in this capacity most likely owed Knight Service directly to the royals.

BENDLET - Here is some information about the black bar or "bend" on the red shield with the lion.  A bend is a broad, diagonal band across the shield representing either a scarf worn like a sash, or the shield suspender of a knight or military commander.  It has often been granted to those who have distinguished themselves as commodores.

"Commodore" is a naval rank used in many navies that is superior to a navy captain, but below a rear admiral. ... As an official rank, a commodore typically commands a flotilla or squadron of ships as part of a larger task force or naval fleet commanded by an admiral. Since some English wars involved battles at sea, and since the Wickendens have a long history as sailors, it is likely that the Knights Service provided to the King and Queen may have involved activity in the navy and service at the rank of a commodore.

A charge half the width of a bend is termed a bendlet. A bend signifies defense or protection, and is a bearing of high honor. As on the Wickenden Arms, the bend is assumed to go from the upper right corner of a shield to the lower left.  This assumes that the orientation of the corners is reversed when viewing the shield from the front.  If the direction of the band is reversed, then it is a bendlet "sinister," which has been used occasionally as a mark of illegitimacy.  This may be why the "owner" of the red shield with the lion and the bendlet going from upper left to lower right when viewing the shield from the front, placed the comment below the image that the Arms were "unofficial," which is presumably what was believed to be the meaning of the designation "illegitimate."

CHEVRONELS - Here is some information about the chevrons on the blue - Azure -  Arms.  The chevron represents the roof of a house, derived from the French word "chevron" meaning rafter. It signifies protection. The chevron was granted to those who had participated in some notable enterprise, had built churches or fortresses, or had accomplished some work requiring faithful service.

The chevronel, is the diminutive of the chevron and is much narrower. Chevronels may be stacked on top of each other or side-by-side at the same height, which is termed interlaced, or braced. So the Wickenden Arms include interlaced chevronels.

Since the Wickendens owed Knights Service to the Priory of Michelham, perhaps they were granted Arms with interlaced chevronels to signify service in building the church or monastery.

KNIGHTS SERVICE - A table is under construction showing the key dates, Wickenden lands for which service was provided, organizations to whom service was provided, monarchs and wars during which service may have been recognized by granting of Arms.  While not yet complete, the table has yielded some interesting suggestions, which have been integrated into the text of the section on Why Wickenden?  The table is attached here in draft form:

LATEST QUESTIONS - It is clear, as indicated on the Why Wickenden? page, that there were several possible opportunities for Wickendens to have been awarded the two arms that are on record in the Burke's General Armory.  However, despite additional research on the Wickenden Coats or Arms, there are several remaining questions, among which are the following:

1. Were these arms registered by the King or Arms at the Ulster Office, and if so, were they granted to Wickendens living north of the River Trent?
2. Did these Wickendens still own Knights Service to the Prior of Michelham, and did Thomas de Wickenden or some other contribute to the construction or extension of the Prior?
3. Did William Wickenden succeed William Fitz-alan as owner of the Manor of Lewisham in Cowden, and is that why his coat of arms includes a similar lion rampant?
4. What exactly is the significant of the bandlet - a diminuitive to the status of the Earl or a diminutive to the role of a Commodore?

UPDATE - Most of the questions posted in earlier replies have been answered, with a number of specifics still unanswered:

1. Questions one, two and three: It appears that the Wickenden Arms were granted by the Ulster Office of the King of Arms because Knights Service to the Priory of Michelham and to the Monarchy - for the Manor of Cowden Lewisham - , were at one point originally provided by the Duke of Norfolk, which county was located "north" - actually south of the river mouth, but north of most of the river - of the River Trent, the boundary of the Office's authority.  Evidence includes a.Both the symbols for the Priory - cross crosslet fitchee - and for the Monarch - lion rampant - are included in the Duke's Coat of Arms and are featured in the two Wickenden Arms; and b. Documents appear to indicate that Thomas de Wickenden provided Knights Service to the Priory and  William Wickenden provided Knights Service to the Monarchy.
2. Question four:  It is possible that the bandlet indicates that William Wickenden served in the Queen's Navy in some capacity below that of a Commodore.  However, since we believe that William resided in Cowden, and documents indicate that the King had given the Manor to William Fitz-alan, prior to its purchase by William Wickenden, it is perhaps more likely that the bandlet indicates a diminution of rank from that of William Fitz-alan, who was the premier Duke and also the Earl of Arundel, while Wickenden was just an Esquire or landed gentry - although the helmut style is also an indication of this rank.
3. The precise meaning of the  bandlet is therefore still an open question, as are the dates of these events and the process through which these Arms were granted to the Wickendens.