Fill or empty fuel tanks in the VAB

To answer #1 I constructed a test rig.

As you described, the outer-engines overheat faster than the inner one regardless of vertical placement. This means that the overheating is not a result of the heat emission map (which extends in a cone underneath the engine that drops off rapidly and is hotter than the engine itself). When I placed temperature gauges on the same rig it didn't overheat at all which inadvertently gives us our answer.

All parts have three config parameters relating to heat: maxTemp, heatConductivity and heatDissapation. By default conductivity and dissapation are set to 0.08 and they are never redefined in the config files. This means that all parts, even the tiny temperature gauge will receive heat from neighboring components at the same rate.

In our scenario the inner engine has four components with which to transfer heat via conduction but the outer ones only have one so they overheat faster. To prevent the outer-engines from overheating you could install a heat sink in the form of a few structural girders.

As for #2 the wiki states that:

As of 0.18.2, this tank does not transfer heat effectively. Engines attached directly to it and running at full throttle (especially Mainsails) will overheat.

I suspect this is a deliberate tweak in the coding since my usual trick of adding a small girder between the tank and engine reduces overheating by an approximate factor of 8 instead of a factor of 2 which you would expect from abusing the heat conductivity parameter as described above.

EDIT

Based on a comment by Philipp I re-checked the results of #1 and it turns out that the inner engine heats up at a slower rate by transferring heat upwards into the command pod, not across to the other engines as I assumed.

Yes, you can save a bit of money by not completely filling your fuel tanks.

You can easily verify this in the VAB when you right-click on a fuel tank and watch how the vehicle cost in the lower left corner changes while you increase and decrease the fuel amount.

The prices for one unit of fuel are:

Liquid fuel: 0.8
Oxidizer:    0.16
Solid fuel:  0.6
Monoprop:    1.2
Xenon:       4.0

The price for fuel is already included in the list-prices of all tanks. The 850 funds price tag of a FL-T400 fuel tank, for example, consists of 666.4 funds for the tank itself, 149 funds for the liquid fuel and 30.6 funds for the oxidizer.

However, reducing the fuel amount is really only economical for solid fuel and xenon where fuel-cost is a very large fraction of total part-cost. Liquid fuel, oxidizer and monopropellant are so cheap compared to their tanks that you pay much less when you just substitute the tank with many smaller ones. Three FL-T100 tanks cost (and weight) less than one 75% full FL-T400.

Exception: When you have a manned command pod but no RCS thrusters on the ship, always remember to remove all the monopropellant from it to save both money and mass.