MySQL Replication Master Slave Architecture - Web app that is based on the SLAVE DB
Yes
It's not going to be a problem but of course it depends on the problem itself. Lets use the term - incident. Problem is already a problem by definition. Solve incidents, analyze the cause , prevent the problem!
In case of an error, try to think what caused it. You are right about worrying, as it is a very common cause of issues.


Try to strictly separate the replicated objects from the unique ones in slave. Different DB
Use some constraints and "tips" for the replication process, like:


--replicate-wild-ignore-table=db_name.tbl_name
include slave status verification in your backup procedures
redirect all reads to the slave (of course the important reads should be done in master and ONLY they)
backup the slave only
include the GTID (you might need it)


Be more AGILE with it. Do it, start testing and you will see how it goes. It's easy and you sound competent enough to do it.

Always monitor your replication. Using GTID and row-based replication may help you a bit with the consistency.