You are right,

Some first level spells are not (that) useful for first level characters

The combat rules for measuring the duration of effects (including spells) state:

When the rules refer to a "full round", they usually mean a span of time from a particular initiative count in one round to the same initiative count in the next round. Effects that last a certain number of rounds end just before the same initiative count that they began on.

^{(emphasis mine)}

Unfortunately for our first level casters, this means that when they cast a spell which lasts only 1 round, it will fizz away immediately before they get a chance to use it.

So, why would something like this even be a Level 1 option?

First of all, there are several exceptions which makes such spells useful even for a first level character:

• Touch spells allow you to "hold the charge" until you successfully hit with them, and give you a "free" touch attack attempt as part of the action of casting the spell.
  So you can attack on the same round you cast them, as well as keep trying to hit for as many following rounds as you need.
• The spell is still active during other combatants' actions on that round - both enemies and allies - this may come up when:
  • Your spell can counter an enemy's action - for example, casting Unbreakable Heart on yourself will protect you from a Fear spell your enemy will cast during his action.
  • You cast the spell to buff an ally (such as Sun Metal cast on your barbarian friend's axe or Unbreakable Heart cast on a confused ally, allowing him to take his turn normally)
  • An enemy exposes himself to your attack of opportunity (your Sun Metal will be active and enhance your damage during his turn)

But, most importantly, some first level spells are mainly useful for casters of higher levels - but it is still better that they are first level spells, because:

• You can access them earlier - for example, Sun Metal may be cast by a 5th level ranger (caster level 2), if it was a second level spell, that ranger will have to be at least 7th level to cast it.
• You can cast them more times per day - sticking with Sun Metal as an example - a 7th level ranger can cast it 2 times per day, instead of just once if it was a second level spell.
• They don't "compete" for spell slots with the more powerful second level spells - at the end of the day, you probably wouldn't want to trade a casting of Brow Gasher or Wilderness Soldiers, just to gain the benefits of Sun Metal - it is not powerful enough to justify being a second level spell, and if it was, it probably would have never been selected by any ranger.

Hope this helps you making sense of it all...

p.s. - I'm not even trying to discuss whether casting Sun Metal is ever the optimal choice for a ranger of any level - you didn't seem to ask about that, and I normally don't care much about such issues myself, so I may not be the best user to tackle that...

Your assumption of RAW is correct.

You would upon scoring a critical hit and them failing their save, paralyze them for one round. The round starts on your initiative count and the paralysis ends just before your following initiative count.

The benefit is that while you yourself aren't deriving a benefit from the paralysis, the targeted creature loses their action for the following round, and it also allows your party members to more easily target the creature while it is paralyzed by the effect of your weapon.

From a GM and a player perspective however, I could see how you yourself would want to derive some form of benefit from an enchanted item such as a Coup de grace weapon, so I could definitely see allowing the paralysis to last from the standard action you made the attack roll with to the attack roll in your next action. This however is up to your GM.

http://www.d20pfsrd.com/gamemastering/combat

Effects that last a certain number of rounds end just before the same initiative count that they began on.