1. Concerned with the borders, stays in Peebles. Resides.
Presumably Re (concerned with) + sides (borders ) = Resides (stays) but how do you account for Peebles.
2. What in music could be represented as sliding. Glissando,
Well the word means a gliding cadence in music so is the whole clue a definition with no word play?
3. It's rumoured tango dance has authority, They say.
I'm assuming this is T (for tango) + an anagram of has but that still leaves me with three letters I can't account for.
4. Siemens staff, but not the number one, all there. Sane
5. One's heard order for starter is a pudding. Sago