Now I can see the images, a few things.There is not much moss, but there is dog lichen and what looks like new thistle growth coming through - either that or some kind of lichen growth, but if it feels prickly, its thistle. If it is thistle, it's stunted and small because it's being cut all the time. The presence of dog lichen is evidence of a compacted, poorly drained lawn which has not had any proper maintenance carried out, and which may frequently be waterlogged. Aeration is essential, even if that's only with a standard garden fork, and applying a moss killer such as Vitax Green Up Lawn Sand should also reduce or kill the dog lichen. RHS information on dog lichen here https://www.rhs.org.uk/advice/profile?PID=414. There are embedded links in there giving advice on spring, summer and autumn maintenance for lawns.
Your lawn might benefit from an application of Verdone 2 (if those are thistles) which is a lawn weedkiller concentrate to be mixed in a can at the appropriate rates. You need to use it a minimum of 4 days after cutting the grass (because of the thistles, a week is better) on a day when there's no rain forecast and the grass blades are relatively dry. You may need to reapply next year - thistles are pretty resilient, but are not great in a lawn for bare feet. I do not recommend the granular, all in one feed, weed and mosskill preparations - the weedkiller element is ineffectual for one thing, and its really easy to under or over dose.
I'm curious as to why the lawn edge in the second and third picture is so much higher than anything which surrounds it - this is not good, because there's a high risk that standing on the edge will break the lawn up, and roots are fully exposed. Is there a reason why its like this, it looks like its been cut back with a half moon edger or similar recently.
It may be that your grass gets waterlogged in winter, or maybe it doesn't - if regular waterlogging occurs, then it may be necessary to install drainage. Either way, the evidence in your lawn means that 'deep watering' is almost the exact opposite of what it requires, though currently, you clearly haven't had a lot of rain, because the dog lichen looks like dry curly plates rather than soggy, bloated flatter ones. Still, where you live, it won't be long before you do get rain, I'm sure. Otherwise, I'm still going to recommend you get a copy of The Lawn Expert by D. G. Hessayon - it will be useful ongoing, but if you do get it, ignore any advice about killing earthworms in the lawn. The products that did this have long since been withdrawn, and earthworms perform valuable functions, so no one kills those any more.
Yes, you can start treating your lawn now, especially given the presence of lichen, so aerate it now. But if you look at the RHS link, under non chemical control, click on 'spring and summer' and it will give some guidance as to what you can do now. The golden rule is never feed the lawn with a spring/summer formulation past August in the UK,and where you are, end of July, first week of August. There are Autumn feeds you can use in September though.
As for the link to the edging, well you could use it, but to be honest, although they seem like a good solution, they're a pain - the grass will overgrow and you won't easily be able to trim it back with 90 degree edging shears. Bear this in mind - if a lawn butts up at the same level to, say, paving, then every year in spring, you take your half moon edger and insert it at the edge of the grass up against the paving, and take out about a half inch in width of the lawn, to depth of about 2 or 3 inches. You do this because otherwise, the lawn will grow over the paving, and you take out that much so you can get your edging shears into that small gap and keep the lawn edge trimmed. That is not possible with this type of edging strip. But if you want to try it, that's your choice.
Because the lawn is not level, but humped up in the middle, a better solution is to remove the lawn altogether, re-level and lay new turf, but that's a lot of work. Alternatively, reset the paving stones in that area a little higher (you'll need to angle them so they butt up with the others around the lawn) and fill in any gap between those and the lawn with some soil, sloped down to the stones, and seed it, then follow the procedure outlined in the paragraph above this for keeping the edges tidy from spring next year.
I just noticed the photo that shows a shrub, with the grass looking a bit grey beneath. All that needs is trimming with your half moon edger, taking out about 3-4 inches of turf to make a sharp edge. The only reason it looks the way it does is because the planting is too close to the lawn edge and it's not getting sufficient light and air.