We want to buy a house in 3 years — are we close?
For your target neighborhood, you'll need about $92k for 20% down plus closing. You have $38k earmarked and you're saving $1,400/mo. At that pace you land at $88k in 36 months — about $4k short. Three options: save $115 more per month, push the timeline to month 39, or accept 18% down and pay PMI for ~14 months.
Let's bump savings by $150/mo.
Updated. New target hit at month 35 with a $2,100 buffer. I've moved the extra into your house-fund bucket at 4.20% APY and adjusted your monthly spending plan to keep the rest of your goals on track.