Sure you can, as long as you meet the prestige class’s prerequisites. The Commoner class does not have any special rules about its advancement, it’s just a particularly weak base class.
It’s actually the easiest way to enter the Survivor prestige class (Savage Species), which requires that one’s highest base save bonus be lower than one’s character level. In a similar case, the Adept is the easiest way to enter the Hexer prestige class (Masters of the Wild), since it requires lightning bolt as a divine spell, and neither clerics nor druids get it, nor is it found on any Domain (amazingly).
I would comment that a common way to handle games where you start as Commoners is to have you “trade in” Commoner levels for PC-class levels, to avoid you taking pretty serious hits to your abilities relative to your nominal ECL. This is a non-issue if your DM knows how to compensate, but it might be worth mentioning as an idea to your DM.
The Beloved of Valarian, as you have noticed, isn’t really intended for Paladins. You can get in (and wind up with both a unicorn and another mount, which could itself be a unicorn), but you don’t progress any of those abilities, which means stagnating your Smite, your spells, and that mount that you’re interested in.
Really, the class is for Rangers to become quasi-paladins, because you can just
Be a Paladin, have a Unicorn, worship Valarian, call yourself Beloved
The primary class feature of the Beloved of Valarian class is the unicorn companion. Well, a Paladin can just have one of those.
In the Dungeon Master’s Guide, page 204:
A paladin of 6th level or higher can use a [...] unicorn [...] as a mount.
So instead of getting a not-unicorn special mount at 5th level, get a unicorn at 6th level. You count as one level lower than your actual paladin level for the purposes of your special mount’s features.
Then worship Valarian, and you’ll effectively be one of the Beloved. A reasonable DM should have no problems with you calling yourself one/being treated like one in-character, either.
This is actually better than what a Beloved of Valarian gets, because your Special Mount improves with level, while the Beloved’s does not aside from the Telepathic Bond and Celestial Charger class features. Paladins instead receive an Empathic Link, though, and the Celestial template doesn’t do much to an already-intelligent unicorn. Despite not having the Celestial template, the unicorn is “from the celestial realms.”
Otherwise, you’re lacking a couple of +2’s to wilderness-y skills; you can easily get these from feats if you desire them. Timelessness, Uncanny Dodge, Nature’s Understanding, and Wood Repulsion aren’t very high-impact features, though the Timelessness is a nice one from a flavor perspective. One solution would be to be a race that gets Timelessness anyway; another could be to just ask the DM for it since it’s not really going to change the game, just change your character’s outlook on life.
The hard ones are ethereal jaunt and mass baleful polymorph. They’re both just once per day, but they’re also both reasonably powerful and fairly flavorful. These you would simply have to accept you aren’t going to get without actually taking the Beloved of Valarian class.
Unless your DM agrees to work with you:
Were I your DM, I’d work with you to build a feat chain for high-level Paladins of Valarian to get these features. Maybe something like these:
Beloved
Requirement: Must be a paladin of Valarian with a unicorn as a Special Mount
Benefit: You gain the Timelessness quality, add Knowledge (nature) and Survival and paladin class skills, and add the following spells to your paladin spell list:
1st: calm animals, charm animal, delay poison, detect snares and pits, entangle, eyes of the avoral, longstrider, pass without trace, speak with animals
Special: If you turn away from the faith of Valarian, you lose this feat and any feats that require it. You regain this feat if you return to the faith (including any atonement that may be necessary).
Because you can’t get the unicorn mount until 6th level, you can’t take this feat any earlier than that, either.
Entrusted
Requirement: Beloved
Benefit: You may, once per day, become ethereal as if by the ethereal jaunt spell cast by a sorcerer of your paladin level.
You also add the following spells to your paladin spell list:
2nd: barkskin, hold animal
3rd: heart’s ease, neutralize poison, remove disease, water walk
Since it requires Beloved, you don’t get this until level 9, the same level as a Beloved of Valarian. Also, you don’t need special rules about sharing the spell with your mount, because you have the Share Spells feature.
Spear of Valarian
Requirements: Entrusted, ability to cast 4th-level Paladin Spells
Benefit: Once per day, you may cast a special version of the baleful polymorph spell that affects all evil creatures within 30 feet of you (as a 10th-level druid). Affected creatures are transformed into harmless Diminutive mammals (rabbits, squirrels, or the like) or Fine vermin (nonthreatening spiders, centipedes, or the like). All affected creatures are transformed into the same type of creature. See the baleful polymorph spell description in the Player’s Handbook for details.
You also add the following spells to your paladin spell list:
4th: blinding beauty, commune with nature, freedom of movement, spear of Valarian
The requirement of 4th-level Paladin Spells puts this to 15th level; a bit earlier than when a Beloved of Valarian gets it, but reasonable enough.
These allow you to be a paladin, while earning some of the special features that Beloveds get in a timely fashion, without making them “freebies.”
Best Answer
Any time you multiclass, you gain nothing but the features of your new class. Your previous classes don't increase. This is the same regardless of which new class you enter.
Sadly, the Core Rulebook does nothing to explain how prestige classes work, just presenting them as extra classes that have prerequisites to enter, but the rules above still apply.
There are, occasionally, some classes that increase existing class features. These classes will tell you when and how it happens. Most commonly, spellcasting prestige classes will effectively increase your position on the spells table of a spellcasting class you possess (and not, for instance, bloodlines, or school powers and arcane bonds).