Do Diablo II mercenaries gain new skills as they level?

Mercenaries do not learn new skills as they level up, but specifically regarding the Act II mercenaries, they do have different skills in different difficulties.

See this page on diablowiki for a list of mercenary skills. An excerpt from the page regarding the Act II mercenaries:

Combat (Normal): Prayer
Combat (Nightmare): Thorns
Combat (Hell): Prayer
Defense (Normal): Defiance
Defense (Nightmare): Holy Freeze
Defense (Hell): Defiance
Offense (Normal): Blessed Aim
Offense (Nightmare): Might
Offense (Hell): Blessed Aim 

@Oak has this completely in the bag. One more thing to keep in mind that is that mercs are often used as an additional set of gear slots for the purpose of auras - there are some incredible higher-level runewords that grant some incredible auras. So keep in mind that you could get a fairly high-level Might or Fanaticism aura from a particular runeword when selecting which merc you want - it can be quite a pain to get a merc up to higher levels.