Can you even mix races in D&D 3.5e? Is it allowed to have a multi-race character?
[RPG] Can you have a multi-race character
dnd-3.5eraces
Related Solutions
Well, Wizards released an entire web enhancement devoted to cats as an April Fools article, which has a lot of cool options for cats. It does have at least one “playable” cat (Cheshire Cat, to be exact), but it’s not really a very playable option because it has LA +3. If it lost the DR, toned down the ability score bonuses a bit, it could possibly pass as a more playable LA +1.
The more typical answer to the desire to play as a cat is to use the tibbit race from Dragon Compendium. These shapeshifters can take on the form of a housecat, and are an LA +0 race (i.e. in line with the regular races). In their regular form, they are Small, have −2 Strength but +2 Dexterity, and Darkvision out to 60 ft.
Catfolk are in Races of the Wild, but they’re fairly disappointing. They have LA +1, and get very little to show for it, sadly.
Shifters from Eberron Campaign Setting or Races of Eberron are quasi-lycanthropes that can “shift” or become more bestial, with fangs or claws or what have you. Could be somewhat feline, but mostly it’s just humanoid with claws etc. They are LA +0, but tend to be fairly underwhelming even for an LA +0 race: while shifting, they’re pretty typical, but that means when they’re not shifting, they’re sub-par.
Both tibbits and catfolk can probably qualify as “feline” for the purposes of the feats and similar in the Fabulous Cats! article. Shifters could if you let them, I suppose.
As for homebrew, bhu has a fair amount of kitty-related homebrew, and he’s a fairly noted ’brewer. I haven’t personally read any of his cat stuff, though.
[RPG] Do races suitable for PCs that have racial bonuses on skills have those skills as class skills
No.
A race defined by class levels with a racial bonus in a skill does not automatically treat it as a class skill. The rules text quoted in the question has been taken out of context and does not apply to PC character creation.
The text referenced in the Monster Manual is written in the context of building and reading monster statblocks. The Skills heading of Reading the Monster Entries not only clarifies this, but also says that having class levels exempts a monster from receiving these free class skills.
Skills
This line gives the creature’s skills, along with each skill’s modifier (including adjustments for ability scores, armor check penalties, and any bonuses from feats or racial traits). All listed skills are class skills, unless the creature has a character class (noted in the entry). A creature’s type and Intelligence score determine the number of skill points it has. (7)
The text quoted on page 307 is a further elaboration on the above text from page 7 meant to explain how to interpret and rebuild a monster's statistics. It mentions skill ranks and racial bonuses because a skill is usually only listed in a monster's entry if that monster has a rank or a racial bonus to that skill. Indeed, the Glossary entry on class skills goes on to explain when you should interpret a skill listing as a class skill or whether the skill is listed simply because it's relevant to the monster's special abilities. This text has nothing to do with PC character creation, and should not be taken out of context to apply to how to build PC characters.
Not only has the text been taken out of context, but also you can easily see plenty of evidence that proves contrary to what this question suggests. For example, there exist numerous examples of NPCs that do not receive free class skills just for having a race with bonuses on skill checks. Some PC races explicitly give class skills as a racial trait.
Best Answer
No
You can't apply multiple races to one character, except under unusual circumstances where some special feature explicitly says you can.
There are specific races called half-orcs and half-elves, for example (the unsaid other half is human), but you can't just be a "1/4 gnome 1/2 elf 1/4 human."
Of course there are some specialized templates, like half-dragon, all of which have rules of their own you'd need to read and understand and have GM approval to use. Same with using any monster race as a character, those have level adjustments and such - see the SRD section on Monsters as Races.
But
OK, having said that, from your other question it doesn't appear that you and your players are playing D&D 3.5e using the printed rules pretty much at all. You can't multi-race, you can't be multi-class at level 1 (you take a single class with a single level, unless you're using the Gestalt rules), and so on. Effectively you're playing Calvinball, not D&D.
And that's totally fine. You don't have to use the rules. Sure, let someone be 1/9 a bunch of races, you do you. But if you aren't using the real D&D, then we can't really help you with questions on this site because the only answer is "make it up yourself." We can deal with questions here that:
But that's about it. If you are trying to learn the real rules, feel free and ask us about them (but please show research effort by reading them first - you don't seem to have read any of the basic character creation rules given these questions). If you are trying to come up with house rules, we can give you balance feedback but you have to use your words and explain them in full.