Do triggers start to degrade in performance (or sometimes fail) on highly transactional tables?
are insert triggers on that table reliable to fire every single time?


Absolutely. Triggers aren't optional, they WILL fire. 

Triggers CAN cause a performance impact on highly transactional tables because the trigger fires synchronously to your DML operation. On an insert trigger, if the trigger execution fails, your insert fails. You won't get confirmation that your insert completed successfully until your trigger has also completed successfully. 

In this way, your insert trigger WILL make your insert more expensive, because the trigger is adding work to a previously simple INSERT statement. Whether it's going to be acceptable performance will depend on what your trigger is doing. 

The same performance problems that can affect any query or procedure can also affect triggers. You'll need to tune them and pay attention to whether that trigger is doing it's job effectively.


  How many transactions per second can SQL Server table triggers handle? 


This isn't a straightforward question. It depends on your schema, your data access patterns, your server hardware, etc. Needless to say, some companies are absolutely running SQL Server at massive scale with transactions per second that are mind boggling. But they have fast hardware, large budgets, and very skilled teams building their systems. 

For this part of the question to be answerable, you'd need to propose a specific scenario &amp; problem to see if anyone can help resolve that more specific issue.