Villain is a reg.

Should I continue on this turn. Is seems to me that turn improves his range and I have showdown value + if he raises it's tough to continue.
Check-back is standard?

River, since he didn't bet, is there any reason for me to bet here. I rarely win with KQ on this board and villain's range seems very weak.
Any reason to turn my hand into bluff?

$0.10/$0.25 No Limit Hold'em Cash, 5 Players
Poker Tools Powered By Holdem Manager - The Ultimate Poker Software Suite. View Hand #36579681

Hero (BTN): $25 (100 bb)
SB: $30.12 (120.5 bb)
BB: $14.11 (56.4 bb)
MP: $25.35 (101.4 bb)
CO: $31.30 (125.2 bb)

Preflop: Hero is BTN with K Q
2 folds, Hero raises to $0.50, SB folds, BB calls $0.25

Flop: ($1.10) 7 9 9 (2 players)
BB checks, Hero bets $0.75, BB calls $0.75

Turn: ($2.60) T (2 players)
BB checks, Hero checks

River: ($2.60) A (2 players)
BB checks, Hero ???

Standard turn barrel imo. You can fold out some small PP, some 7x and a lot of Ace highs + you have very good equity vs villains range.