I hope it is not rude to sort of just show up and ask a million questions, but I really do appreciate any advice!
Welcome to the forum, Mike. We don't mind questions at all, and you're asking the right ones.
I've owned my 152 since 2001, and rented it from 1995 when I started taking lessons in it.
I'm in Denver (BJC), but with no illusions about flying a 150/152 into the mountains. My experience is in a 152. With that said, would DA be significantly more of an issue in a 150? The "mission" is me just flying around and occasionally my wife and I (~340# together) going on short hops together.
I'm a flat-lander, so I'll leave it to someone else to answer your question about flying in a high DA area.
Is budgeting $100/month for a tiedown reasonable? Is an uncovered tiedown a terrible idea?
Call the local FBO/airport and ask about the tiedown cost. At my airport, tiedown is free, my enclosed hangar costs $100/month. At airports back East, even a tiedown costs more than $100/month. The problem with uncovered tiedown is weather. In Denver, you'll be clearing snow from the plane often. In Kansas, the biggest problem is hail storms.
Is budgeting $1,200/year for insurance reasonable?
You'll be surprised that insurance isn't as high as you think. The biggest factor is what you want to insure the plane for (hull value). Most 150-152s are insured for $20K-$25K. The other factors are the size of your home airport (busy or not), grass or concrete runways, pilot's time-in-type, etc. Most of us pay no more than $800/year for insurance. Some pay as low as $400/year.
Is budgeting $1,000/year for the annual and $1,500/year for unplanned maintenance reasonable?
Those numbers are reasonable. In the early years of ownership, the annual cost tend to be higher because you might have/want to fix things that the previous owner put off.
Is budgeting $500 for a pre-buy inspection reasonable?
That sounds reasonable.
How do you recommend thinking about an engine overhaul reserve? My first thought was the overhaul cost divided by (TBO minus SMOH) but I wonder if that is overly conservative.
Overhauling the engine in a 152 at a good national/regional shop will cost $18K to $20K. A local mechanic might do it for $12K-$14K. I think the best option is to buy a plane with about 1000 hours left before TBO. If you fly about 100 hrs/yr, it'll be 10 years before you have to "face the music".
Is, say, $2,000 on hand after the purchase a reasonable amount in case something breaks early that needs immediate repair?
That's a good number. Likely, you'll find somethings that you want to add/improve after buying the plane: new carpet, new seat covering, intercom, strobes, etc.
Are there big budget areas I'm missing other than the purchase price, down payment, interest rate, acquisition cost (prebuy, travel, etc.), tiedown/hangar, insurance, annual, unplanned maintenance, fuel cost, oil cost, and engine reserve?
Some states/municipals have property tax or registration fee on airplanes. In Kansas, property tax on an airplane amounts to about 5% of the book value ($1000/yr on a $20K airplane). I had to pay that for a while until my plane became older than 30 yrs and is exempted.