[RPG] Do Arcane Ward’s starting hit points include the INT modifier once or twice

The ward begins at maximum hit-points. The first version of the PHB stated the same passage as:

The ward has hit points equal to twice your wizard level +your Intelligence modifier.

I believe the change in text was made to address questions about the maximum, but in doing so I guess it introduced this question.

Yes it is viable but not with Shield.

Your best bet would to use Mage Armor, since it is a single action spell as opposed to the reaction of Shield that has specifics on when you are allowed to cast it (emphasis mine).

Casting Time: 1 reaction, which you take when you are hit by an attack or targeted by the magic missile spell

So unless you actually have someone hit you repeatedly you can't cast this spell all that often per PHB 196. At least by RAW.

A reaction is an instant response to a trigger of some kind...

This section goes on to say that the opportunity attack is one such instance of a reaction but you can't take it unless something would trigger that as detailed in all the descriptions of reactions. This why when you ready an action you are required to specify a trigger.

Mage Armor could be used as often as necessary to regenerate the ward, unless there is errata somewhere that changes the wording to slots as opposed to level. One thing to note is that spells with verbal components could be overheard and thus could give away position or spoil the element of surprise, so choose when to do this with care.