Potions of Healing are available in the PHB and are listed as a "common" magic item in the DMG alongside 1st level scrolls.
1. does this imply that potions of healing should be widely made available for pcs to purchase?
I would say that is the implication, yes. Not so much because they're Common items, but because they're listed in the PHB equipment list. Of course, the DM can always rule otherwise.
2. If so; do potions of healing become the new CLW wand of this edition?
Doubtful. First, the
potion is more than three times as expensive per hit point as the
wand, in an edition where there are a lot more options for spending your money--it's not just about buying gear any more. Second, you need 47
potions to equal the healing potential of one
wand; that's 23.5 pounds of
potions (imagine lugging around three one-gallon jugs of milk everywhere), whereas the 3E
wand was stated to weigh about 1 ounce. Third, just because you can buy a
potion or three doesn't mean the local herbalist has 47 of them waiting to go. My experience may not be typical, but I've known few DMs who wouldn't balk at such a request.
3. Should (or do) 1st level scrolls or other common magical items share the same availability as potions of healing?
I would say no, but could be persuaded to change that ruling. My argument is that the
potion is in the PHB and the other items are not, implying that the
potion is more widely available than other items. The logical counter-argument would be that the PHB has limited space and chose to list the magic item that is most widely useful, and the DMG suggests a price point of 50 gp for common items, equivalent to what the
potion costs.