Hello! I'm experienced Linux user for decades, but maybe I'm doing something wrong in modern times. Please give advice.

I installed on CentOS 7, x64 platform: "uname -a"

Linux whatever.whereever.eu 3.10.0-514.6.1.el7.x86_64 #1 SMP Wed Jan 18 13:06:36 UTC 2017 x86_64 x86_64 x86_64 GNU/Linux

did follow the steps:

# unzip traccar-linux-3.10.zip
Archive:  traccar-linux-3.10.zip
  inflating: traccar.run
  inflating: README.txt

# ./traccar.run
Verifying archive integrity... All good.
...
Service traccar installed
...

/opt/traccar/bin/installDaemon.sh

...
Service traccar installed

/opt/traccar/bin/startDaemon.sh

...
Service traccar started

cat /opt/traccar/logs/wrapper.log.20170301

WARNING|wrapper|Service traccar|17-03-01 00:26:36|YAJSW: yajsw-stable-12.08
...
FINEST|1297/0|Service traccar|17-03-01 00:26:40|
FINEST|1297/0|Service traccar|17-03-01 00:26:40|INFO 3/1/17 12:26 AM: liquibase: Successfully released change log lock
FINEST|1297/0|Service traccar|17-03-01 00:26:40|[main] INFO org.eclipse.jetty.util.log - Logging initialized @4286ms
FINEST|1297/0|Service traccar|17-03-01 00:26:41|[main] INFO org.eclipse.jetty.server.Server - jetty-9.2.21.v20170120
FINEST|1297/0|Service traccar|17-03-01 00:26:42|[main] INFO org.eclipse.jetty.server.handler.ContextHandler - Started o.e.j.s.ServletContextHandler@74834afd{/api,null,AVAILABLE}
FINEST|1297/0|Service traccar|17-03-01 00:26:42|[main] INFO org.eclipse.jetty.server.ServerConnector - Started ServerConnector@415ef4d8{HTTP/1.1}{0.0.0.0:8082}
FINEST|1297/0|Service traccar|17-03-01 00:26:42|[main] INFO org.eclipse.jetty.server.Server - Started @5610ms
FINEST|1297/0|Service traccar|17-03-01 00:27:27|[qtp45880026-45] WARN org.eclipse.jetty.servlet.ServletHandler - /api/socket
FINEST|1297/0|Service traccar|17-03-01 00:27:27|java.lang.NullPointerException
FINEST|1297/0|Service traccar|17-03-01 00:27:27|        at org.traccar.api.AsyncSocketServlet$1.createWebSocket(AsyncSocketServlet.java:37)
FINEST|1297/0|Service traccar|17-03-01 00:27:27|        at org.eclipse.jetty.websocket.server.WebSocketServerFactory.acceptWebSocket(WebSocketServerFactory.java:166)
FINEST|1297/0|Service traccar|17-03-01 00:27:27|        at org.eclipse.jetty.websocket.server.WebSocketServerFactory.acceptWebSocket(WebSocketServerFactory.java:153)
FINEST|1297/0|Service traccar|17-03-01 00:27:27|        at org.eclipse.jetty.websocket.servlet.WebSocketServlet.service(WebSocketServlet.java:151)
FINEST|1297/0|Service traccar|17-03-01 00:27:27|        at javax.servlet.http.HttpServlet.service(HttpServlet.java:790)
FINEST|1297/0|Service traccar|17-03-01 00:27:27|        at org.eclipse.jetty.servlet.ServletHolder.handle(ServletHolder.java:812)
FINEST|1297/0|Service traccar|17-03-01 00:27:27|        at org.eclipse.jetty.servlet.ServletHandler.doHandle(ServletHandler.java:587)
FINEST|1297/0|Service traccar|17-03-01 00:27:27|        at org.eclipse.jetty.server.session.SessionHandler.doHandle(SessionHandler.java:221)
...
...
FINEST|1297/0|Service traccar|17-03-01 00:28:49|        at org.eclipse.jetty.util.thread.QueuedThreadPool$3.run(QueuedThreadPool.java:555)
FINEST|1297/0|Service traccar|17-03-01 00:28:49|        at java.lang.Thread.run(Thread.java:745)

cat /opt/traccar/logs/tracker-server.log

2017-03-01 00:26:37  INFO: Operating system name: Linux version: 3.10.0-514.6.1.el7.x86_64 architecture: amd64
2017-03-01 00:26:37  INFO: Java runtime name: OpenJDK 64-Bit Server VM vendor: Oracle Corporation version: 25.121-b13
2017-03-01 00:26:37  INFO: Memory limit heap: 409mb non-heap: 0mb
2017-03-01 00:26:37  INFO: Character encoding: UTF-8 charset: UTF-8
2017-03-01 00:26:37  INFO: Version: 3.10-SNAPSHOT
2017-03-01 00:26:41  INFO: Starting server...
2017-03-01 00:31:27  INFO: [3A270DA8] connected
2017-03-01 00:31:27 DEBUG: [3A270DA8: 5001 < 217.71.46.14] HEX: 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
2017-03-01 00:32:03  INFO: [3A270DA8] disconnected

...as you see, I've installed meanwhile also iphone traccar client. It decodes to:

GET /?id=123456&timestamp=1488321009&lat=59.488121&lon=24.861747&speed=0&bearing=0&altitude=19.9838&batt=49 HTTP/1.1
Host: whereever.whatever.eu:5001
Accept: */*
Accept-Language: et-ee
Connection: keep-alive
Accept-Encoding: gzip, deflate
User-Agent: TraccarClient/4.0 CFNetwork/808.3 Darwin/16.3.0

And from webgui part, logged in, enabled one new device with code 123456. And waiting, but nothing happens? PS. Please reply.

Logged in to web no, registerd device from Traccar clinet on OS and still nothing happens.
besides log gets:

2017-03-01 00:30:09  INFO: [6BE56B5C] connected
2017-03-01 00:30:09 DEBUG: [6BE56B5C: 5001 < 217.71.46.14] HEX: 16030100c5010000c1030358b5f9f192478f77630b48b15eb75270d63d68eb253972e3a348ac9b38ccd54800002600ffc02cc02bc024c023c00ac009c030c02fc028c027c014c013009d009c003d003c0035002f0100007200000012001000000d686f6d652e6164686f632e6565000a00080006001700180019000b00020100000d0012001004010201050106010403020305030603337400000010001b001908737064792f332e3106737064792f3308687474702f312e310005000501000000000012000000170000
2017-03-01 00:30:55  INFO: [6BE56B5C] disconnected

and nothing visible still from GUI (id 123456)

Done something wrong? :-S

BR.,
Liivo

Please next time don't paste massive log files straight into the post. If you really think it's useful to provide, which in this case just the same exception repeated over and over, upload it somewhere and provide a link.

As for your issue, you are using wrong port. Why have you decided to change it from the correct default port (5055)?