Pega doesn't have specific preferred tools and vendors, but here are some suggestions based on how other customers have approached this.
Here are some responses
1) Recommendations to test SOAP/TEST API's
ANS - See Mary Ann's recommendations above but there are many options in this space.
2) Need recommendations on testing Network and Page Latencies.
For testing service or application performance, tools such as BlazeMeter (or JMeter which is the open source solution), Neoload (load testing) and many others can help in this. You can also look into vendors such as Tricentis Tosca suite of test tools that have a comprehensive approach to testing that covers much of regression testing as well as performance testing.
3) Services or integration tools for measuring the performance in production and proactively mitigating them.
Production monitoring can also be achieved through various approaches. You should first consider using Pega Predictive Diagnostic Cloud that can monitor alerts, exceptions and other metrics that matter from the application perspective. If you would like to monitor other aspects of the environment such as the app or database server and other aspects of the of the deployment stack, there are many solutions out there such as AppDynamics, DataDog, NewRelic and so on. Which one to use might depends on your particular deployment setup, for example cloud vs on prem, and other variables. Perhaps your company might already have some that is being used for other applications.
4) Offline mode in case one of the nodes go down without asking the users to switch or logoff.
5) Making all Pega applications highly available 99.99%.
One additional recommendation I can make is to work with a Pega partner such as Valori who have expertise in developing a comprehensive testing solution for customers on Pega and they can help you with working out a good strategy for all of this.
I agree with Linoy and Mary Ann on their answers there are a lot of tools out there that can help you out with the points you mention.
At Valori we have a lot of experience with testing Pega and we have worked with these tools on numerous projects helping customers getting the feedback they need.
Which tool fits your situation best, is difficult to say based on the shared information. Maybe we can get in contact directly to get a better understanding of your situation and see if we can help you out.
SoapUI is great tool for testing any SOAP services and doesn't require much work to make it running.
Automatrix team is using JMeter for performance testing of mobile offline service. With Charles proxy IP set on device's WiFi we can grab all the communication requests. In JMeter we set GET/POST HTTP_Requests for login page first to fetch session cookies then grabbed service requests are performed in a loop. If request needs to be customized or responses parsed you can use JSR Processors and write groovy script for it.