It seems a little bizarre that the journal for which I am reviewing a paper doesn't show line numbers... is it just me or is this incredibly unhelpful? Or rather, is there something I am missing here... some easy way to refer to a line of text without having to count the paragraphs or quote whole sentences...?

I think I am getting a little grumpy in the sweltering heat!

I should add... it's like a 40 page review so unless I am missing a trick this is going to be a bit of a nightmare when it comes to referring to certain places.

Your text editor might be able to add those line numbers for you.
Otherwise I'd send it back to the editor to get them added.

Ask the editor if there was a small mistake or tell him that you cannot be as thorough without suitable referencing options.
Don't waste your time reviewing the paper in detail if they can't be bothered to add the line numbers.

Thank you. Have emailed. Will see what response is. I am part way through it now but would prefer it if they could add page numbers. Man it is hot today.

Also noteworthy, it isn't blinded, as in I can see the authors' names. Never had that before but I just checked and this journal doesn't operate double blinded reviews. So the reviewer can see the author but they can't see us. I find it odd somehow, even though it is the journal's policy. It isn't influencing me but it is weird that I actually know the authors for sure (often you can guess anyway but still).

In my fields, I've mostly seen single-blind reviews and usually see the authors. It doesn't influence me, I just focus on the material at hand. Although, sometimes, on viewing a profile after I've done the review it can be surprising that experienced researchers can miss obvious things.

That's interesting. Most in my field (at least the ones I am familiar with) are double blind. I've finished it now, and yes, it didn't make a difference. Although it was interesting. I'm discovering that some people seem to submit more draft versions of a manuscript as opposed to highly polished ones. Maybe it is to get a feel if the manuscript would be accepted in a particular journal with revisions. Or maybe it is using the peer review process more as a useful earlier stage of writing feedback mechanism rather than as a final gate-keeping plus feedback one (which is how I have tended to see it).

That might explain why you notice that experienced researchers miss important things.