As you can see in this table, there is no conflict between Full backup and Log backup. They can occur at same time, and, when log backup completes, log will not be cleaned until full backup (or diff backup) completes.
Not all administrative tasks are allowed to run concurrently. In the table below, a black circle indicates two operations that cannot run in a database at the same time.
References:
Concurrent Administrative Operations
Seeing that the recovery model is set to simple and msdn states that the simple recovery model does not support point-in-time recovery - Does this mean that I won't be able to use my transaction log backups to restore the database in a disaster to an hour before it happened?
Even taking a transaction log backup is not supported for databases using the SIMPLE recovery model. This is a restriction of the database engine based on how this recovery model works, and the recovery features it doesn't support, as you mentioned.
A transaction log backup maintenance plan task automatically skips databases in SIMPLE recovery to avoid causing errors.
Which backup should be done first, the database backup or the transaction log backups? Articles that I'm busy reading say I should do the database backup first and then the transaction log backup else I will get maintenance plan errors, but I'm currently first backing up my transaction logs and then data databases and I'm not getting any errors.
For the reasons I mentioned above, it won't matter for databases using SIMPLE recovery, as they will be skipped by the transaction log backup task.
For databases in the other two recovery models, a full backup must exist before you start taking transaction log backups (just the first time), or you will get an error -- this is probably what the articles refer to.
Point-in-time recovery ability is normally driven by business need -- in other words, you determine how critical the data is and how much you can afford to lose, then set the appropriate recovery model to meet those needs, and finally create a backup solution.
Even though SIMPLE recovery does not support point-in-time recovery, if an hour of data loss is okay, perhaps a differential backup solution could work for you. (There are far too many variables that go into developing this kind of solution to give you a complete picture with what was provided in the question.)
Best Answer
Working out which Log Backup to restore is all based on LSN's. You have to apply all log backups since the full (or differential, but you didn't mention them) backup, including log backups taken while the full backup was in progress.
There is no danger in trying to restore a log backup out of sequence, it just won't work. Kin answered above how to find the actual log backups you need to restore.
It's important that you have a continuous chain of log backup files. If you are missing one, or if it's corrupt then you can't restore anything after that point.