Can I extend my stay in Thailand with a 60-day visa, and what are the rules for visa extensions and border crossings?

First of all, there is no such rule "two 30 day visa extensions per year".

The rule you may have mistaken is, "30-days visa exemption are eligible enter Thailand through land border only twice a calendar year".

So, these are not counted as "visa-exempt through land border",

- enter by air

- enter on pre-purchased visa

- enter on re-entry permit

30 days tourism extension could be applied for visa-exempt entries and tourist visa entries.

Q1. Does traveling with a 60 day visa affect this extension from the UK?

-> No. You can apply for tourism extension on SETV entry no problem

Q2. can you travel on a 60 day visa more than once per year?

-> Logically, no limit on entering Thailand on SETV. But it's until the immigration officer at passport control upon your entry Thailand decides you are not a genuine tourist.

Q3. does the 30 day no visa border crossing affect the two extensions?

-> It doesn't affect tourism extension on visa-exempt entry, but it IS counted as one of twice per year of visa-exempt entry by land.

Q4. can you fly in and out of Thailand and not affect the extension.

-> no, you can apply for tourism extension on visa-exempt entry or SETV entry by air. But it's another story if the immigration officer at passport control on your entry point would let you enter or not.

Attached is the announcement about limit of twice per year on visa-exempt entry by land