First, to answer your 3 questions:
- Yes, you can safely make a 50mm deep cut with a handheld router, but only if you take several passes, only cutting part of the depth at a time. If you try to make the entire cut in one pass, you'll very likely either damage the bit, stall your router, injure yourself, or some combination of these.
- There are certainly bottom-bearing bits this long, but it seems to be harder to find top-bearing (pattern) bits that will make that deep a cut. However, you can buy a collet extension which will increase your maximum depth of cut. Keep in mind that a pattern bit's cutters don't need to be long enough to cut the entire depth all at once. For your first pass, the bearing rides along your pattern, then for deeper passes the cut from one of your previous passes becomes the pattern (assuming your bearing is the same diameter as the cutter--which isn't always the case).
- If you are asking whether this will be faster than making several passes with a 3/4" dado stack, then the answer is probably no.
The main reason you cannot find a dado set that cuts a 50mm wide dado is that no consumer-grade table saw has an arbor long enough to accommodate that width, at least in the US (and my understanding is that dado blades are prohibited as unsafe on European table saws). Such a wide dado stack would probably also strain or stall the motors of many consumer-grade table saws. In the US, radial arm saws were also commonly used with dado blades, but have declined in popularity. If you want to use a dado stack or wobble dado blade, you'll have to make multiple passes.
There are several options available for cutting this with a router, from setting up guides that register against the router's baseplate and using a large straight bit, to a spiral bit with a bushing and pattern, to a straight or spiral pattern bit. You can buy large spiral pattern bits, but they're expensive. You can buy a collet extension if your bit isn't long enough to cut 50mm deep. Regardless of which bit you use, you will still need to make several passes and depending on your setup you may be just as well off hogging out the bulk of material with some other tool (even a bandsaw, drill press, hand saw, jigsaw, table saw, or circular saw) before cleaning it up with the router.
Of these two overall options, the table saw will still probably be the fastest. I'd suggest ganging up several boards, clamping them together (in a crosscut sled, if using a table saw), and cutting the dadoes in all of them at once.
Personally, if I was building landscaping projects from pressure-treated wood, I wouldn't worry about cleaning up the joints too much and would just use construction screws that are designed not to corrode when used with modern pressure-treated lumber. Depending on the application, I might not even bother to use glue.