You should take a look at MySQL error logs.
If you dont have access to machine (hosting etc) you may ask your administrator or helpdesk for that log.
MySQL supports very big queries. Im not sure if there is any limit, but when you are using network - you may have problem with packet size.
You may check --max_allowed_packet
in MySQL configuration, and try to set bigger packet size. Im not sure about default configuration, but it may be 1MB which may be too small value to get query with 7000 updates at once.
MySQL may need more RAM to process query like this.
If you cant reconfigure MySQL you have to split your big query to smaller queries somehow.
You may also read this for more information:
devshed - MySQL server has gone away
You asked:
Any possibility to run multiple queries even faster?
there is no simple answer for that question. It depends on query, database schema etc.
Increasing MySQL cache size in configuration file may help a lot in most cases related with big simple updates with not much computing, because database engine will operate on RAM memory, not on hard disk. When big cache is used - sometimes first big query may be slower, because data is not yet loaded into RAM, but when it finally loads - queries that need a lot of read/write operations will work much faster.
Added later:
I assume your data processing needs php deserialize() function which may be hard to implement in pure SQL and you have to do it in PHP :) If you have access to server console you may create cron (linux sheduler) job, that call PHP script from shell during night.
Added later later
After discussion in comments i have one more idea. You can make full database or one table backup from phpmyadmin, download it, restore data on home computer (on Windows you may use XAMPP, WAMP server). On your home computer you can run mysql.exe and process data locally.