Would never recommend any form of plunger, almost inevitable that you are going to get a face full of something. Except a set of drain rods, because then you can stand well back
This, definitely. A plunger is likely to either fail to remove the blockage, spray you with filth or both. If you're going to plunge, you need to be able the ensure a seal between the plunger and the bowl and you're not going to get that with a mop head.
If you're going the mechanical route, a toilet auger is probably your best bet. Like pretty much anything, the right tool for the job makes it much easier. Drain rods for a drain, like bitslice says, but a toilet auger is better if the blockage is actually in the toilet. Although the blockage might well not be in the toilet - the first noticable result of a blockage in the drain is often a blocked (or partially blocked) toilet.
You could also locate the drain inspection cover (outisde, probably in your garden/yard near the outlet from the bathroom) and lift it to see if there's any sign of a blockage. If there is, it will not be pleasant. A lot of...stuff...can backlog there before the blockage causes symptoms in the toilet.
It might be easier to just call a pro to do the job.
Last thing - if you use any chemicals to try to unblock it and then call a pro out to do the job, tell them what you used. Even if it's just washing up liquid it's probably worth mentioning it, but definitely tell them if it's anything else (like the sodium hydroxide you've already tried). If you put chemicals down there and it's still blocked, those chemicals may well still be there and may pose a risk to the person unblocking the toilet.