On my server I have observed that whenever I teleport to a far location or even spawn/respawn in Spawn Chunks, the map takes some seconds to load and you fall in the void for bit or see missing chunk here and there, which is really unaesthetic. I'd like to overcome this, so I tried adding named mobs (Silverfish) to the places I wanted to TP and behind the spawn as well, just to be sure. By the way, it didn't produce any observable result and te problem still occur. This bothers me a bit, because I knew that spawn chunks are constantly loaded. Might it be a client-side/connection lag (my ping is 45ms, looks fine) or am I getting something wrong?
Unfortunately, this is largely unavoidable. When going to a chunk that your client does not have loaded, the server will have to send it to the client. This takes a nonzero amount of time, even if the server has the chunk loaded already. Note that the server's spawn chunks (not your spawn/bed location, but the server spawn) are always loaded, so if you experience the same issue when dying/respawning at server spawn, loading the chunks will not help.
A faster server or having a faster connection between your server and your client* would mitigate, but not entirely remove the problem. If you do not experience the same kind of problems when respawning at the server spawn, then loading might help. You can force chunks to be loaded by the server using vanilla chunk loading mechanics. This requires an unbroken line of loaded chunks from spawn with hoppers pointing into them - this youtube video seems to be the most referenced design (plus an explanation of how it works). It basically involves using hoppers to keep the chunks loaded, and requires an unbroken chain of loaded chunks between spawn and where you want to be loaded.
*Your ping only measures the round trip time. You need to be concerned about the actual connection speed, or how many bits/second can be sent.
I don't know how you can fix falling into the void randomly, but if some chunks aren't loading and others are you can press F3 + A and it will reload all of the chunks.