Hi,
If you switched from a student visa to FLR(M), then you should reach 5 years on the expiry date of your visa (June 8th)
Do you have a record of your previous visa (the first FLR(M) from when you switched)? A photocopy or anything?
The start date on that visa will be the first day of your 5 years. Since the FLR(M) visas are issued for 2.5 years exactly, the expiry date of your current visa should be 5 years from the start date of your first FLR(M).
So, if your current visa expires on June 8th 2018, you should reach 5 years on June 8th 2018. You can apply for ILR up to 28 days before this, so I think that's May 11th 2018. That means you can apply any time between May 11th and June 8th.
Do you think I would be safe applying in that first week of June? Also, if I decided to mail the application (rather than go in person) would that week be an appropriate time to send it?
As long as it is postmarked on or before June 8th, you will be fine... so I would send it at least a couple of days before that.
Also a quick question about the timing of the LIUK Test. Can you take that anytime or does it have to be close to your application date? For instance I know other documents have to be signed/dated within 28 days of applying - is there a similar restriction of when you have to pass the test?
No, it can be taken at any time as it never expires. You'd be best taking it as soon as you can to get it out of the way and make sure you have it done in time (you could have taken it at any time since you moved to the UK).
Just make sure you keep your pass certificate safe as you will need it for UK citizenship as well, if you decide to apply.