In Texas, who is responsible for ensuring that outdoor pipes are weatherized- the tenant, or the landlord?

You should first take a look at your lease and see if it addresses maintenance responsibilities. Generally, if you rent a single family home you are responsible for the simple maintenance of the property which this certainly sounds like it would fall under, the Landlord is responsible for the home mechanicals functioning as they should (the boiler works, the pipes or roof don't leak, etc.) In a multifamily rental simple maintenance is not so straight forward. Anything which could be considered a common area would probably fall under the responsibility of the Landlord anything in an area clearly under your control would be your responsibility. I wouldn't recommend asking the general public do you legal research for you though.

In most every case the landlord is responsible for things outside of your rental. You are responsible for things inside or under you control, and anything you have agreed to handle.

Insulation on pipes would normally be the landlords responsibility, not the tenants. If you are concerned write the landlord a nice letter. Inform him that you noticed some of the outdoor pipes are not insulated, and you wanted to bring it to his attention so the pipes don't burst.

It is possible that freezing pipes never occur in parts of TX.

Yes, but what if they burst. The renter is responsible for all damages. The landlord gave the home to the person in good condition so the landlord is not liable for insulation unless the tenant bring it up before the damage occurs.

The law isn't going to address this issue specifically. You will only find references to tenant and landlord responsibilities in general. Since you don't want my opinion, I won't provide it.

I'm sure that would be the landlord. I am a landlord and I would never expect a tenant to do that