When you're running your phone in its Safe Mode, only the base Android operating system gets loaded into memory during the start up process. No third-party apps or services that typically get preloaded too are excluded. While in Safe Mode you'll notice most things you do are a bit sluggish when first used, as everything that typically gets preloaded into RAM is now starting up as a new process/service. If there is some third-party app you previously installed is interfering with your text messaging app in some way, it is less likely to be able to while in Safe Mode. The issue being, now there's a much more clear indicator that is the problem to address but no, Safe Mode isn't a magic way to specifically tell you what app is the source of the problem. You still need to deduce just what app you added before that's the problem.
Given the sporadic and random pattern of your description of the problem, I tend to doubt this is caused by another app but it's worth trying to eliminate the possibility it is. Using Safe Mode costs nothing and it's just a temporary thing to use as a diagnostic task.
You've referred to multiple phone restarting that occasionally and temporarily solves your problem but at some point realize that isn't a really practical fix. Restarting your phone is a good way to clear up something that's more of a one-off glitch, but it won't fix something like a corrupted settings file. That's still something you need to take care of in some kind of direct manner. The notifications issue 'could' be some setup matter in the Notifications menu in either the Settings menu or the text messaging app itself (more likely the phone's Settings menu than the app's Settings menu given you're tried different text messaging apps); the old and new messages not showing properly as bold and normal is likely to be a text messaging app matter but does that particular problem happen in all the text messaging apps you've tried? If yes that would indicate something with your text messaging service -- assuming you're using your cellular service phone number. If no, have you tried clearing the cache and/or data for whatever text messaging app you're using as your default? Note that clearing the app's data will wipe all its settings/options files, essentially returning it back to its original, first time used state so you'll have to set up the app from scratch again, and be sure backup up your text messages as you might have to restore them. This isn't as drastic as a Factory Reset as it's more targeted to just your text messaging service (with a Factory Reset you have to restore your phone's entire user account data, a more comprehensive matter.)