$10k is less than 10% of $125k, so there's almost no point of putting all that down.
Either put 3% down on an FHA, or put 5% down on an 80/15 conventional combo. Or better yet, do an 80/20 combo and put nothing down.
Points or not, expect to put $1k earnest money down upfront (refunded at closing) and to pay normal reasonable upfront closing costs like appraisal and home inspection.
"No points" btw usually means slightly higher rate. Be advised that $125k is a small loan in today's mortgage world, so don't expect brokers to beat down your door. If you have excellent credit and can document acceptable income, you might just try applying at your bank.