Wiring new basement with DSL/Phone
Yes, wire connectors, or multiple wires per screw terminal, are common for telephone connections.  It's unlikely the wiring you have will cause problems for your xDSL signal.  I wouldn't suggest re-configuring it until you try to activate your service and determine if you get satisfactory speed.  That way, you have a baseline reference before adding any jacks for your basement.
If the cable is CAT5e and connectors are RJ45, you could even use those for hard-wired Ethernet with the appropriate connectors.  In a larger home, that can be an advantage for multiple wireless access points, hard-wiring media streaming devices &amp; game systems, etc.
For the above reason, you might think about re-terminating the cables with RJ45 connectors if you ever decide to put an Ethernet switch where the cables are simply wire-nutted together today.  You can wire up a small RJ45 patch panel as an ad-hoc splitter for continued telephone use (if anyone ever uses those again) or there are RJ45 splitters.