After a bit of searching I answered several of my own questions:

1. Banks and in person transactions only. (Similar to the US before the USG figured out the internet was a thing 10-15 years ago. You may be able to work with an investment firm online, though I'm not sure on this.)
2. You can find auction dates and historical data on the internet.
3. ??? (I'm pretty sure foreigners can purchase bonds but this could be complicated by your visa status.)
4. ??? (My guess is "No." I think you would need a Philippine bank account to initiate a securities purchase.)
5. ??? (I believe US anti-double taxation laws would apply for how this income is taxed. If the Philippine rate for interest income is lower than the US rate you would pay the Philippines tax and whatever remaining percentage to the US tax rate. Example: If the Philippines tax rate on the interest income was 5% and the US tax rate on that same income was 15% you would pay 5% to the BIR and then pay the remaining 10% to the IRS, no matter what you pay at least whatever the minimum US tax rate would be. If the Philippine tax rate is over 15%, say 20%, you would pay nothing to the IRS, the full 20% would go to the Philippines. However, the Overseas Income Exclusion tax credit, aka FEIE (foreign earned income exclusion), may apply if you have spent 330 days outside of US territory...this would be prorated if you spent less than 330 days outside of the US. This gets very complicated very quickly, hire a CPA.)
6. Yes they do, but depends on the bank. The PHG officially states they offer 4 week bills, but bank's websites seem to suggest the minimum is 12 weeks (also, a minimum P100k bond purchase).
7. No, of course not. It's the Philippines after all. Why would they do anything that makes life easier?

 

This is the first sign you have noticed? You should pay more attention.