Whether these are valid, and what they should be used or not could be a massive answer and be mostly opinion. So I've tried to answer this as objectively as possible, from the point of view of whether they are (or look like they are) being actively maintained/updated. Scroll to the bottom for the summary.
SQLDiag is still in use for SQL Server 2016, but is possible up for deprecation
This utility may be changed, and applications or scripts that rely on its command line arguments or behavior may not work correctly in future releases.
SQL Nexus looks like it is still being developed, and the most recent version (22.214.171.124) was released on CodePLex
PAL tool the most recent version (2.7.7) was released last May (can't find further information on the life of this, I'll carry on looking - it does appear that Data Collector most of what this does).
So yes, although lots of these features have been included in the Data Collector component which can partially replace lots of what these do component (partially discussed here. ), coupled with a Management Data Warehouse, these also makes it easier to manage multiple servers.
Database Tuning Adviser is packaged with 2016 and appears to still be a core component. As far as alternatives go, you could check things yourself using DMVs, and there are third party tools like Qure Analyzer (I have no experience on this tool and I only mention it as an example, if you want to know more check out this SO post which asks for alternatives to Tuning Adviser
BPA doesn't appear to be maintained, although I can't actually find anything confirming that it's officially got rid of (it's actually a Windows Server tool, which complicates finding info, and there is a BPA for Windows Server 2012) - there's no BPA for SQL Server 2014 or later. So it might still be useful if you're on 2012 or earlier. This SQL Server Central thread discusses this, but there doesn't seem to be a clear replacement that does it all (I'll add to this if I can find something).
Policy based management is still a thing, take a look at - Central Management Server, which helps you manage multiple servers (coupled with data collectors to replace some elements of the other tools).
Yes, most of these tools can still be used, but some are ageing and may not be updated.
SQLDiag is packaged in 2016 (but looks iffy for the future, SQL Nexus and PAL tool have both got version released on codeplex within the last year. Database Tuning Adviser still seems to be a core component of SQL Server. Best Practice Analyzer hasn't had a released for SQL Server since SQL Server 2012, doesn't appear to have an official word but doesn't look like it's being developed. Policy based management is still in use, and looking at Central Management Server with Data Collector may replace, in part, some of the other tools for ongoing management.