Electrical – New sub panel wire sizes

Well, I would not say you're doing anything stupid. You have some very good questions.

First,

• If you did decide to direct bury the wire the minimum depth for direct burial is 24", not 18".
• At least three #6 ( black, red, white) and one #10 insulated ground ( green coating).
• Anything <= 60amps just requires a #10 insulated ground with green coating.

Second, Consider voltage drop:

• Load: 30 Amps @ 240V Single Phase.
• Length of run: 80 feet
• Wire Size: #6 Copper
• Voltage: 240V
• Voltage Drop: .81 %
• Voltage At End of Circuit: 238.05

That is less than 3% which the NEC recommends for a feeder. Very Good !

Note: I would hesitate to install the ground rod if you have a in-ground pool in line of the transformer.

Correction: This being a feeder would require a grounding electrode!

You are undone. You have 130A of loads which are quite conceivably on at the same time, so you need #1 or #0 wire. I would err on the high side, honestly. I'd even think about 2/0.

You also need four conductors unless the conduit is all-metal in which case you need three.

1" conduit will only support #4 wire (85A) in four conductors... Or #2 wire (115A) in three conductors**. And even then, you're at conduit limits, which makes for a rather miserable pull... for which you'll need special tools and skill not lending itself to DIY.

If this is a DIY, then dollar for dollar, it'd be far cheaper to install 2" conduit so you can easily and legally pull what you need.

** 130A is allowed with 90C terminations, but good luck finding those... and you'd really want to run that by your local inspector before buying anything. I mean you do expect to occasionally run the wire at actual full capacity, literally, people would burn their fingers if they touched the conduit. Seems a bit crazy.