#1
I'm trying to intonate my Tele, and the strings on it are not what you'd call a new strings. I'm wondering (because I'm having troubles with the first few frets being sharp), does it matter a lot if the strings are not new? Or should you definitely get a new set of strings before you do any action on the nut, truss rod, bridge...
#2
Yes - old strings can make the strings intonate significantly flat, so it's a waste of time setting the intonation with old strings, because it'll be completely wrong when you replace them.
Rig Winter 2017:

Fender Jazzmaster/Yamaha SG1000
Boss TU-3, DS-2, CS-3, EHX small stone, Danelectro delay
Laney VC30-112 with G12H30 speaker, or Session Rockette 30 for smaller gigs
Elixir Nanoweb 11-49 strings, Dunlop Jazz III XL picks
Shure SM57 mic in front of the amp
#3
Definitely. You should always check the intonation and setup when changing strings. And you should always change strings if the intonation and setup seem way off.
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#4
You do always want to intonate for the strings that are on the guitar, so getting a new set. However, if it's a 3-saddle style bridge, you might not be able to get it perfect as they aren't the easier to work with