Why is hydrogen ion much more likely to be +1 than -1?

Firstly, hydrogen doesn't really belong to the group of the Alkali Metals and is increasingly placed separately from the rest of the PT, like in this example. Historically it's often been placed in Group 1 because of its half-filled $1text{s}^1$ orbital.

Secondly, in chemistry, hydrogen doesn't really form 'naked' protons. Instead, in aqueous media so-called oxonium ions ($text{H}_3text{O}^+$) are formed. Here's the oxidation half-reaction, schematically:

$$2text{H}_2text{O}(l)+text{H}_2(g)to 2text{H}_3text{O}^+(aq)+2e^-$$

The equilibrium:

$$ text{H}_3text{O}^+(aq)to text{H}_2text{O}(l)+text{H}^+(aq)$$

leans extremely to the left: aqueous solutions contain almost no protons, only solvated ones.

Using Lewis notation oxonium cations have the structure:

These are tetrahedral in geometrical shape, with the oxygen atom at the 'top' of the tetrahedron and a full $text{2p}_{x,y,z}^2$ orbital (doublet) sitting on top of that.

'Higher' oxonium ions like $text{H}_5text{O}_2^+$ and $text{H}_7text{O}_3^+$ also occur.

Thirdly, hydrogen can also form anions, aka hydride ions; $text{H}^-$, which are quite stable in hydrides like $text{LiH}$, $text{NaH}$ and $text{KH}$.

Why does hydrogen belong in the alkali metals/group 1 and not in the halogens/group 17?

The electronic structure of the halogens is very different from that of hydrogen. The valence electron structure of the outer (valence) electrons of Group 17 is that of one filled $text{s}^2$, two filled $p$ orbitals (e.g. $p_x^2$ and $p_y^2$) and one half-filled one (e.g. $p_z^1$) (seven electrons in total). This leads to high electronegativity down the entire group with the elements trying to acquire an electron to complete the half-filled $p$ and obtain the octet structure.

Hydrogen however, in the oxidation state $text{+1}$, forms covalent bonds that through electron pair sharing mimic a full $text{s}^2$ molecular orbital (see the oxonium ion).

The title question, for these reasons, doesn't really make a lot of sense because whether hydrogen gets in the $+1$ or $-1$ oxidation state depends on chemical context.

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As regards your 'Nonessential add-on': hexane and hexafluoro hexane are homologous because in these compounds hydrogen and fluorine have the same valence ($text{-1}$).