The multiclassing rules in Chapter 6 of the PHB cover this.

Can spell slots from the the total number available from the Multiclass Caster table (PHB pg165) be used to cast either known Bard spells or prepared Cleric spells?

Yes. A Bard 4 / Cleric 4 can cast four first level spells and three second level spells per day (and a few higher level spells). These could be all cleric spells, all bard spells, or a mix between the two. You make the choice as you cast the spells (as always).

Note that multiclass characters will often have spell slots whose level is higher than any spell that character can prepare. You can use these slots by casting spells at a higher level. For example, our Bard 4 / Cleric 4 could cast Cure Wounds as a 4th level spell, but couldn't prepare Banishment.

Is it correct that Cleric spells must still be prepared each day from the set of available Cleric spells for the character's level?

Yes. Page 164 (under the Spells Known and Prepared heading) spells this out pretty clearly:

You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class.

Is it correct that Cleric spells must use Wis as the primary stat, and Bard spells must use Chr?

Yes. Again, page 164 spells it out:

Each spell you know and prepare is associated with one of your classes, and you use the spellcasting ability of that class when you cast the spell.

No Official Answer

Unearthed Arcana is playtest material, so there are bound to be mistakes. The page you linked to, The Faithful, was subsequently updated in Wizard Revised. The next article in the series, A Trio of Subclasses has a link where they were collecting feedback on Wizard Revised, but it is expired.

I didn't see anything in the revised version that answers your question. Getting an official answer to your question will have to wait until it actually becomes part of an officially published product, or they release another revision and ask for feedback again.

Advice

Considering that the latest version explicitly states using your Wizard Spell Save DC where other saves are required, I would also substitute Intelligence for Wisdom and Wizard level for Cleric level in any powers granted through the subclass. Despite a theological background, the character is still a Wizard. In fact, one of the later abilities mentions coming earlier than clerics get it because of the character's method of studying to learn it.