Is crond (cron server) running?

Use the pgrep or ps command as follows to verify that crond is running:

pgrep cron
ps aux | grep cron
sudo service cron status
sudo status cron

Sample outputs:

Jan 17 17:43:21 planetvenus cron[229]: (CRON) INFO (pidfile fd = 3)
Jan 17 17:43:21 planetvenus cron[240]: (CRON) STARTUP (fork ok)
Jan 17 17:43:21 planetvenus cron[240]: (CRON) INFO (Running @reboot jobs)
Jan 17 18:01:01 planetvenus cron[240]: (*system*cache) NOT A REGULAR FILE (/etc/cron.d/cache)
Jan 17 18:01:01 planetvenus cron[240]: (*system*output) NOT A REGULAR FILE (/etc/cron.d/output)
Jan 17 18:11:45 planetvenus root: cronplanet
Jan 17 18:12:11 planetvenus root: cronplanet
Jan 17 18:17:01 planetvenus CRON[3911]: (root) CMD (   cd / && run-parts --report /etc/cron.hourly)
Jan 17 18:23:31 planetvenus cron[3927]: (CRON) INFO (pidfile fd = 3)

Where are cron logs stored on Ubuntu Linux?

The logs are stored in /var/log/cron.log file. You need to configured it as follows. Edit /etc/rsyslog.d/50-default.conf file using a text editor such as vi or nano:
$ sudo vi /etc/rsyslog.d/50-default.conf
OR
$ sudo nano /etc/rsyslog.d/50-default.conf
Find the line:

#cron.*                          /var/log/cron.log

Uncoment the line (i.e. remove #):

cron.*                          /var/log/cron.log

Restart the following two services:
$ sudo service rsyslog restart
$ sudo service cron restart
Sample outputs:

rsyslog stop/waiting
rsyslog start/running, process 4004
cron stop/waiting
cron start/running, process 4014

Now, where are cron errors logged in Ubuntu LTS server?

You need to view /var/log/cron.log file using the grep or tail command:
$ sudo grep something /var/log/cron.log
$ sudo more /var/log/cron.log
$ sudo tail -f /var/log/cron.log
$ sudo egrep -i 'error|log' /var/log/cron.log
$ sudo tail -F /var/log/cron.log
Sample outputs from my log file:

Jan 17 18:29:27 planetvenus cron[3988]: (CRON) INFO (Skipping @reboot jobs -- not system startup)
Jan 17 18:34:25 planetvenus cron[4013]: (CRON) INFO (pidfile fd = 3)
Jan 17 18:34:25 planetvenus cron[4014]: (CRON) STARTUP (fork ok)
Jan 17 18:34:25 planetvenus cron[4014]: (CRON) INFO (Skipping @reboot jobs -- not system startup)
Jan 17 18:45:01 planetvenus CRON[4039]: (root) CMD (/root/scripts/mk.rss.feed)
Jan 17 18:45:02 planetvenus CRON[4038]: (CRON) info (No MTA installed, discarding output)
Jan 17 18:50:01 planetvenus CRON[4051]: (root) CMD (/root/scripts/mk.rss.feed)
Jan 17 18:50:03 planetvenus CRON[4050]: (CRON) info (No MTA installed, discarding output)
Jan 17 18:55:01 planetvenus CRON[4060]: (root) CMD (/root/scripts/mk.rss.feed)
Jan 17 18:55:02 planetvenus CRON[4059]: (CRON) info (No MTA installed, discarding output)