The elf racial trait Silent Hunter says;
Silent Hunter: Elves are renowned for their subtlety and skill. Elves with this racial trait reduce the penalty for using Stealth while moving by 5 and can make Stealth checks while running at a –20 penalty (this number includes the penalty reduction from this racial trait). This racial trait replaces elven magic.
And the level 10 ability of Unchained Stealth says:
10 Ranks: Stealth check penalties for moving quickly are halved, including the ability unlocked at 5 ranks, moving full speed, and reaching concealment after creating a distraction.
So which one applies first? Do you halve the penalties and then reduce them by 5, or do you reduce the penalties by 5 and then halve them?
unless there is a specific rule i don't know about how to solve this kind of things i would go chronologically.
the elven racial trait, not to mix with basic trait (race), is something the character got from being an elf. like low light vision and medium size. the moment he is born (or an adult etc) he has it.
this skill unlock is something one must work hard to learn (gain at least 10 levels before even being able to take it, also be an unchained rogue or take a feat).
so i would say he first adjust by what he always had. then apply what he learn. -5 (or -20 when running) then half rounded up
this would be a logical scaling of abilities when you compere how to advance in ability.:
- elf (or not) but not a 'silent hunter' and no skill unlock - normal penalty.
- silent hunter elf but no skill unlock - lowered penalty for being born talented.
- silent hunter and skill unlocked - the lowered penalty is halved.
if it was the other way around it would have started by unlocking the skill then turning into an elf.
and when comparing two silent hunter elves one with skill unlock and one without the one with the skill unlock would somewhat be panelized for being a silent elf by not giving him the full benefit of the skill unlock.
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For moving at full speed it is indifferent:
"When moving at a speed greater than half but less than your normal speed, you take a –5 penalty."
(-5+5)/2=0
-5/2+5=2.5, but you can't get a bonus by negating a penalty. The ability reduces the penalty, so, at most, it can reduce it to 0.
For the Stealth check while running it is different. A common person can't try stealth checks while running.
The ability says:
"can make Stealth checks while running at a –20 penalty (this number includes the penalty reduction from this racial trait)."
So, the penalty is -20 and the racial trait is already factored in.
The skill unlock affects the penalty, i.e. the -20, reducing it to -10.