Wand with Melee Spell for noncaster

Question

In a DND 5e campaign, my party has received a Wand of Shocking Grasp. It's currently in the hands of a non-caster. Shocking Grasp is a touch melee spell. I have 2 questions:

1. When the spell is cast, it's done as a melee attack. Does the user apply any bonus if they're not a caster class? The non-caster does not have a spell attack bonus but I could see adding a DEX bonus because you're attempting to reach out and touch your opponent, which takes some dextrous targeting.
2. Could the wand be considered a "light melee weapon" allowing the user to use two-weapon fighting along with, say, a dagger? Wands are not classified as weapons, but the definition of "Light" is "A light weapon is small and easy to handle, making it ideal for use when Fighting with two Weapons." which fits the wand.

Answer 1

Since "Wand of Shocking Grasp" is not an item in the DMG or any other supplement I'm aware of, you'll have to ask your DM how it works; we can only speculate. (You should learn what you need to know about how the item works when you use identify on it.)

In general, items that specifically cast a spell but can be used by non-casters, which includes most of the wands, will specify a fixed attack roll or saving throw DC (as appropriate for the spell in question). As an example:

Circlet of Blasting

While wearing this circlet, you can use an action to cast the scorching ray spell with it. When you make the spell's attacks, you do so with an attack bonus of +5. [...]

There is no question as to what the roll should be: It's stated right in the item description.

If the item uses the wielder's spellcasting modifiers, as in the cast of most magic staff items, it will require the item to be attuned to a spellcaster or be used by a spellcaster, so the question is moot.

In the case of a character who can use the item despite not being a spellcaster, there's a specific rule found in the Activating an Item section (DMG p.114):

Spells

Some magic items allow the user to cast a spell from the item. [...] A magic item, such as certain staffs, may require you to use your own spellcasting ability when you cast a spell from the item. [...] If you don't have a spellcasting ability - perhaps you're a rogue with the Use Magic Device feature - your spellcasting ability modifier is +0 for the item, and your proficiency bonus does apply.

A wand is not a weapon and does not use the Attack action, so it's not going to qualify for two-weapon fighting or any other attack-related abilities.

Answer 2

There is no Wand of Shocking Grasp in the DMG. Judging from the other wands there that replicate an offensive spell, some can be used by anyone (e.g. Wand of Magic Missiles) others require attunement by a spellcaster (e.g. Wand of Paralysis, Wand of Polymorph). Let's assume the non-caster can use the wand: if not, end of story.

Shocking Grasp is a cantrip with touch range, and says

Lightning springs from your hand to deliver a shock to a creature you try to touch. Make a melee spell attack against the target. You have advantage on the Attack roll if the target is wearing armor made of metal. On a hit, the target takes 1d8 lightning damage, and it can't take Reactions until the start of its next turn.

1. Assuming your wand works like the spell, then the person using the wand would use an action to use the wand, and then would make a melee spell attack, with their free hand, which uses their proficiency bonus and spellcasting ability modifier as attack bonus. If they have no spellcasting ability, then their spellcasting ability modifier is +0 for the item. (Thanks to Pyrotechnical for this piece.)

   The rule on spell attack rolls states (PHB, p. 206; emphasis added):

   Some spells require the caster to make an attack roll to determine whether the spell effect hits the intended target. Your attack bonus with a spell attack equals your spellcasting ability modifier + your proficiency bonus.

2. Unless the wand has different explicit ruling, the wand itself is not a weapon, it is a magic item that allows you to cast a spell. The attack is done with your hand as a touch attack, not with the wand. So the wand would not be usable as an off-hand weapon.