First, the usual permission bits for the /tmp directory are 1777. The leading 1 indicates that the directory's sticky bit has been set, which means that although multiple users may write to the directory, they retain ownership of any files written there so that only they (or root) may delete them. Hence without sudo you will only be able to delete your own files from /tmp. As well, some files in /tmp may be open and/or locked by other processes and you should not attempt to delete them unless you know the locks are 'stale' - you can check for open files with lsof e.g.

$ sudo lsof +d /tmp

to see any open files at the top level of the directory, or

$ sudo lsof +D /tmp

to show open files anywhere in /tmp and its subdirectories - see the OUTPUT section of the lsof manpages man lsof for an explanation of what the various columns mean.

The second issue you are likely facing is that although commands like rm can take multiple filename arguments (including expanded wildcards or shell globs), there is a limit to the allowed length of the argument list. For example, if we create 100,000 files in /tmp

# touch /tmp/file{000000..099999}
# ls /tmp/file* | wc -l
100000

(so far so good) but if we add a further 100,000 files, the expanded shell glob (i.e. the list of files to be deleted) becomes too long for the ls command line

# touch /tmp/file{100000..199999}
# ls /tmp/file* | wc -l
-bash: /bin/ls: Argument list too long
0

and similarly we can't delete in one go with rm

# rm -f /tmp/file*
-bash: /bin/rm: Argument list too long

The solution is to break down the wildcard list into manageable subsets - you did that manually, but it is possible to do it automatically using the xargs command e.g.

# echo /tmp/file* | xargs rm -f

You could also have used the find command with a -exec or -execdir action and the {} + argument replacement syntax - from man find

-exec command {} +
              This  variant  of the -exec action runs the specified command on
              the selected files, but the command line is built  by  appending
              each  selected file name at the end; the total number of invoca-
              tions of the command will  be  much  less  than  the  number  of
              matched  files.   The command line is built in much the same way
              that xargs builds its command lines.  Only one instance of  `{}'
              is  allowed  within the command.  The command is executed in the
              starting directory.

for example

# find /tmp -maxdepth 1 -name 'file*' -type f -exec rm -f {} +

although the xargs version will likely be faster if you only need to delete from the top level of /tmp (i.e. not from any subdirectories).

If the directory continues to fill up you should really investigate which users / processes are creating the files and why. If you changed /tmp's permissions you should reset them to 1777.

Your current logs are fine, still, those without .1. That is good, and you can remove it with:

sudo rm /var/log/*.1

Now you command doesn't work because of this:

sudo 'Everything here runs as root' > Everything here run as user

So if you wanted to do what you tried the correct would be:

sudo sh -c "echo '' > kern.log.1"

This is because the pipe opens a shell with the current user.