I admire youth, strength and optimism.
If you can get this tree without laying out cash for truck rental, or help, then it might be worth the try. It is a pretty ambitious project. The larger the tree in the ground, the lower the probability of getting enough of the roots to have it survive.
As to which species, it definitely looks like Japanese black pine, which is good, they are well adapted to the Los Angeles climate. It could also be Pinus radiata - the Monterey Pine. Care for both species is similar, so identity is not too important. The owner might know, ask when you go to dig.
Pinus thunbergia 'Thunderhead' is a cultivar of Japanese black pine (JBP) noted for heavy, thick new growth, long needles and being difficult to get to ramify into fine branch patterns. It tends to always look coarse. It has a bad track record for those trying to make bonsai with it. But if you go for a large scale bonsai, it possibly can be made to work. Think finished size of 4 feet by 4 feet. I have never seen a "good" bonsai using 'Thunderhead', the best were long needled and coarse branching. But never say never, it could work, at least you have a fully developed trunk heading into the project.
When to dig - in cool weather before the heat of summer kicks in.
Roots - get as large a root ball as possible. Have lumber on hand at home, to build a grow box of sufficient size to hold the roots. Build props or trellis structures to hold the tree upright, or "in position", (not necessarily upright, this is bonsai) so that there is zero wiggle in the trunk when the wind blows. The tree needs to be stable to build a new root system. Try not to bare root the tree. Near to the trunk, finer roots are the most important. Any root going straight down more than 6 or 8 inches will need to be cut don't keep longer downward roots. Long horizontal roots are a matter for practical consideration. Get them if you can get the fine roots at the end. A long horizontal root with no fine feeder roots might just as well be pruned short to the trunk. Wrap the root ball, with as much native soil as practical, do not bare root the tree. If you can get 50% of the roots, you have done well.
Have your potting media on hand in sufficient quantity before you head out to pick up the tree. One of the good media to use is 100% pumice, sifted to retain right around 3/16th through 5/16th inch size, Basically 1/4 inch particles. No fines, no real coarse particles. The more uniform the particle size the better root development. Of course, perfection is the enemy of good enough, but seriously, do sift your media. I would use 100% pumice, but there are a dozen other media components you can use.
Foliage - definitely prune no more than 50% of the foliage, less would be better. Leave long, even foot long or longer stubs to become deadwood features. Do not trim off branches flush to the trunk until you have time (a year or more) to contemplate styles and see what survives.
Do not worry about temperatures as long as you don't have hard freezes, the tree will be fine. It is good to maybe 29 F with the damaged, newly transplanted root system. Best to avoid freezing if you can move it around, if you do plant it in the ground, drape it with burlap or a tarp when frost threatens.
It will need at least morning sun, and for the first year, it will need afternoon shade during the heat of the day. Site the tree on the east side of the house or east side of a fence or garage or other structure to provide afternoon shade.
Mist foliage daily, even several times a day, with a light spray to just get the needles wet. Thoroughly water the potting media when needed, initially media might not need water daily, then as roots begin to develop, the need to water more frequently will increase.
Hope that helps.
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