Mining bases: mostly you can't. Major civs are very reluctant to sell them, to the point where the only way you can reasonably expect to trade for one is as part of a peace treaty, and sometimes an alliance. Bases belonging to minor civs can't be bought, period. Often the best way to go about it is to bribe someone else to attack the base's owner, and hope it gets killed. Keep some constructors in the area to take advantage of the opening.

Max population - try to keep it below 20b on large planets, 12-14 on medium planets, 8 on anything PQ 10 or below. Even though I play a very influence-heavy game, I don't build influence buildings. Stock markets and total population is where you get your real power.

Influence calculations are absurdly complicated; I looked into them at one point and got hit by a dozen or more factors I could barely identify, let alone explain. In general, though, you should do the following:

1) Build your population. The bigger your civilization, the more influence each planet puts out.

2) Farm bonuses. Mining bases are a no-brainer, but you should also be buying techs with passive bonuses on them, collecting anomalies, and picking up colonization bonuses on new planets.

3) Influence bombs. Once you've mastered 1 and 2, obtain a planet in enemy territory. Invade a minor civ, invade a third civ's planet in your target's area, encourage wars and buy freshly-invaded planets (anything under 1b is fair game). As a last resort build multiple influence bases at once, Ignore their ring of effect; their power drops off absurdly fast with distance, so build them all touching the target planet. Once you get the first planet the rest generally domino for you. In TA, make sure to get the MCC; in DL and DA avoid it like the plague if you actually want to flip planets (but the econ bonus is sweet, so in DL/DA it's a must-have for anything but an influence-based game)

Yearly votes are controlled by the total number of influence points you generate each turn (based on the sum of your planets' influence output) and summed for the year. You can buy and sell these points, but the number of them in circulation is so high the possibility of buying enough to affect a vote is not practical.

So much whitespace...

I don't remember, it's either the generic 8-week window, or twice that (16 weeks).  I'm pretty sure it's 8, though.

Proximity is irrelevant, but there is a civ-wide cap, of sorts.  But before we get into that, open up your Civ Manager screen and view the bonuses-sometimes it doesn't properly register.  Then try adjusting the tax rate.

Mining bases give civ-wide morale, which is capped at 100 after depreciation by population.  Basically what this means is if you have a planet at 16B pop (HW with no farms), you get a -48% modifier applied to all morale sources for the planet.  This means you need 193 racial morale to hit the cap, so if you have say a total of 200 and add in another 50 from a mining base, you will see no difference, on a 16B planet.

For reference:

Population: malus, multiplier, morale to hit cap (100) after depreciation, morale to balance out pop malus after depreciation (no taxation included)

8B: -18%, 0.82, 122, 22
10B: -25%, 0.75, 134, 34
11B: -28%, 0.72, 139, 39
12B: -32%, 0.68, 148, 48
13B: -36%, 0.64, 157, 57
14B: -40%, 0.60, 167, 67
15B: -44%, 0.56, 179, 79
16B: -48%, 0.52, 192, 92
17B: -52%, 0.48, 209, 109
18B: -57%, 0.43, 233, 133
19B: -61%, 0.39, 256, 156
20B: -66%, 0.34, 295, 195
21B: -70%, 0.30, 334, 234
22B: -75%, 0.25, 400, 300
23B: -80%, 0.20, 500, 400
24B: -85%, 0.15, 667, 557
25B+: -90%, 0.10, 1000, 900

Also, these numbers apply for DA/TA; I can't recall if DL had a morale cap (pretty sure it didn't, actually), and while the depreciation numbers are similar, to a point, DL bottoms out at 20% instead of 10% for 25B and up.