ok i'm having weird multiple problems all of a sudden. Monday It started, I noticed a misfire, engine shaking when ever I start my HCHI. it lasts all of 10 seconds then check engine light comes on. also when this started happening at low rpms in traffic my heat turns cold. my engine temp guage stays steady in the middle when warmed up. once on the highway I get full heat till I hit traffic again and the heat gets cold. i'm not loosing any coolant and the coolant level is fine. both problems happened the same day. the codes are p0300, p0303, p0301, p0304. every time I start the car I get the same codes all over again it shakes for 10 seconds then runs smooth all new plugs are in and look great. I know those codes are miss fire on cyl 3,1,4 and 0300 is multiple misfire. what the heck can no heat and misfire have in common. i'm not loosing a drop of coolant and no white smoke out exhaust so i'm ruling out head gasket.
any ideas?

 

Rough running:
1) "new plugs" = recently replaced OEM style Denso or NGK?
2) Bad coil packs? I recently had to replace one on my HCH2.
3) do you ever get a check engine light that's flashing?

Heat:
1) confirm radiator and heater core hoses are actually hot when you're losing heat at idle.

Could there be a vaccum leak issue?