Are primary carbocations the most stable because they have more hydrogens?

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Carbocation stability is about the positive charge, not the number of hydrogens. The positive carbon has only six valence electrons and is electron-deficient. Anything that donates electron density toward it spreads out the positive charge and stabilises the ion.

Alkyl groups do this in two ways. Through the inductive effect (+I), they push electron density through σ bonds toward the positive carbon. Through hyperconjugation, the C–H σ bonds on carbons next to the positive carbon overlap with its empty p orbital, delocalising the charge. More alkyl groups means more of both effects.

So stability rises with the number of alkyl groups on the positive carbon: (CH₃)₃C⁺ (tertiary, 9 α-hydrogens available for hyperconjugation) > (CH₃)₂CH⁺ (secondary, 6) > CH₃CH₂⁺ (primary, 3) > CH₃⁺ (methyl, 0). The hydrogens that matter are those on the neighbouring carbons, and a tertiary cation has the most of them. Hydrogens attached directly to the positive carbon give no hyperconjugation at all.

This order explains why tertiary alkyl halides react fastest by the SN1 mechanism, and why carbocations often rearrange by hydride or methyl shifts to put the charge on a more substituted carbon. Note that resonance can be even more important than alkyl effects: allyl and benzyl cations are strongly stabilised because their charge is delocalised over a π system.