Pipe running into engine block

No. Should be smooth.

if I say possible servo booster is faulty,surely you will feel it on the brake pedal ( stiff ),or the engine running would be erratic.the other alternative situation is the pipe( hose ) which is connected with inlet manifold and servo booster and inside has the check valve probably is faulty. Test : off engine,pump brake pedal several times till gets stiff. Keeping your foot on brake pedal start the engine and if your servo works fine then the pedal goes a bit down,if not will not move at all. If the engine is running erraticly then with engine off, remove the pipe connecting manifold and servo block the pipe you remove from servo side,block it and start the engine,If engine run fine,then you must confirm whether is the servo or the check valve hose. For the check valve hose how to check,just put in your mouth the servo side pipe and blow into,if the air goes in the engine means that the check valve is broken and you must renew the pipe with check valve,if on blowing into will not allow air to get through,means is ok and the servo is the fault. Good luck.

The condensate drain hole or pipe must be blocked. This drains the water to outside the car.

Firstly before commencing any new work , ensure that you do not have an air lock caused by replacing the previous various parts. Follow the pipe from the thermostat, with engine running and it should heat up when thermostat opens. Follow this pipe towards both the Heater matrix and the carburettor. Both should warm up slowly. Have the heating on inside the car to blow warm. Then feel the exit pipes from the Heater Matrix. The exit pipe should also heat up. If they remain cold the you have an blockage within the Matrix which can either be air or debris. To cure air, let car cool down and remove expansion vessel cap. Then remove exit pipe from heater matrix and hold pipe aas high as possible.....if air is causing the blockage then it will purge itself, when water starts coming out of Matrix, reconnect pipe and top up levels as required. If nothing comes out, the matrix is blocked, may respond to after market flushing treatments or may require removal (pig of a job) Do check before this that oil is not getting into water circuit (dark floating bits if mild, all light brown if bad)or vice a versa (white scum on bottom of dipstick)which would indicate blown head gasket is causing car to overheat!

Water Pump REMOVAL & INSTALLATION CAUTION
Never open, service or drain the radiator or cooling system when hot; serious burns can occur from the steam and hot coolant. Also, when draining engine coolant, keep in mind that cats and dogs are attracted to ethylene glycol antifreeze and could drink any that is left in an uncovered container or in puddles on the ground. This will prove fatal in sufficient quantities. Always drain coolant into a sealable container. Coolant should be reused unless it is contaminated or is several years old. 2.4L Engine

1. Disconnect the negative battery cable
2. Detach the Oxygen (O2) sensor electrical connector.
3. Properly drain the engine coolant into a suitable container. Remove the heater hose from the thermostat housing for a more complete coolant drain.
4. Remove upper exhaust manifold heat shield.
5. Remove the bolt that attaches the exhaust manifold brace to the manifold.
6. Remove the lower exhaust manifold heat shield.
7. Break the manifold-to-exhaust pipe spring loaded bolts loose using a 13mm box wrench.
8. Raise and safely support the vehicle. NOTE: It is necessary to relieve the spring pressure from 1 bolt prior to removing the second bolt. If the spring pressure is not relieved, it will cause the exhaust pipe to twist and bind up the bolt as it is removed.
   
9. Unfasten the two radiator outlet pipe-to-water pump cover bolts.
10. Remove the manifold to exhaust pipe bolts from the exhaust pipe flange as follows:
    1. Unscrew either bolt clockwise 4 turns.
    2. Remove the other bolt.
    3. Remove the first bolt. NOTE: DO NOT rotate the flex coupling more than 4°or damage may occur.
       
11. Pull down and back on the exhaust pipe to disengage it from the exhaust manifold bolts.
12. Remove the radiator outlet pipe from the oil pan and transaxle. Leave the lower radiator hose attached and pull down on the outlet pipe to remove it from the water pump. Leave the radiator outlet pipe hang.
13. Carefully lower the vehicle.
14. Unfasten the exhaust manifold-to-cylinder head retaining nuts, then remove the exhaust manifold, seals and gaskets.
15. Remove the front timing chain cover and the chain tensioner. For details, please refer to the procedure located later in this section.
16. Unfasten the water pump-to-cylinder block bolts. Remove the water pump-to-timing chain housing nuts. Remove the water pump and cover mounting bolts and nuts. Remove the water pump and cover as an assembly, then separate the two pieces. Fig. 1: Water pump and cover mounting - 2.4L engine
    To install:
    
17. Thoroughly clean and dry all mounting surfaces, bolts and bolt holes. Using a new gasket, install the water pump to the cover and tighten the bolts finger-tight.
18. Lubricate the splines of the water pump with clean grease and install the assembly to the engine using new gaskets. Install the mounting bolts and nuts finger-tight.
19. Lubricate the radiator outlet pipe O-ring with antifreeze and slide the pipe onto the water pump cover. Install the bolts finger-tight.
20. With all gaps closed, tighten the bolts, in the following sequence, to the proper values:
    1. Pump assembly-to-chain housing nuts:19 ft. lbs. (26 Nm).
    2. Pump cover-to-pump assembly: 106 inch lbs. (12 Nm).
    3. Cover-to-block, bottom bolt first: 19 ft. lbs. (26 Nm).
    4. Radiator outlet pipe assembly-to-pump cover: 125 inch lbs. (14 Nm).
21. Using new gaskets, install the exhaust manifold. Make sure to following the tightening sequence and torque specifications given in the exhaust manifold procedure located in this section.
22. Raise and safely support the vehicle.
23. Index the exhaust manifold bolts into the exhaust pipe flange.
24. Connect the exhaust pipe to the manifold. Install the exhaust pipe flange bolts evenly and gradually to avoid binding. Turn the bolts in until fully seated.
25. Connect the radiator outlet pipe to the transaxle and oil pan.
26. Install the timing chain tensioner and front cover.
27. Install the lower heat shield.
28. Carefully lower the vehicle.
29. Fasten the bolt that attaches the exhaust manifold brace to the manifold.
30. Tighten the manifold-to-exhaust pipe nuts to specification.
31. Install the upper heat shield.
32. Attach the oxygen sensor connector.
33. Fill the radiator with coolant until it comes out the heater hose outlet at the thermostat housing. Then connect the heater hose. Leave the radiator cap off.
34. Connect the negative battery cable, then start the engine. Run the vehicle until the thermostat opens, fill the radiator and recovery tank to their proper levels, then turn the engine OFF.
35. Once the vehicle has cooled, recheck the coolant level, then install the radiator cap.

3.1L Engine

1. Disconnect the negative battery cable.
2. Properly drain the cooling system into a suitable container.
3. Loosen, but do not remove, the water pump pulley bolts.
4. Remove the serpentine belt.
5. Unfasten the water pump pulley bolts, then remove the pulley.
6. Remove the five water pump mounting bolts, then remove the water pump. To install: Fig. 2: View of the water pump mounting and bolt tightening specifications - 3.1L engine shown
   
   
7. Clean all the gasket surfaces completely.
8. Apply a thin bead of sealer around the outside edge of the water pump along the gasket sealing area, then install the gasket onto the water pump.
9. Position the water pump on the engine, then tighten the water pump mounting bolts to 89 inch lbs. (10 Nm).
10. Install the water pump pulley and finger-tighten the pulley bolts.
11. Install the serpentine belt.
12. Tighten the water pump pulley bolts to 18 ft. lbs. (25 Nm).
13. Fill the cooling system.
14. Connect the negative battery cable, then start the engine. Let the engine run until it reaches normal operating temperature, then check for leaks and coolant level. Add coolant, if necessary, then turn the engine OFF.

I had this same problem in my 02 745i... It's leaking from the internal coolant pipe... a $.05 rubber ring is going to cost you 3-5k easily.

You will have an 'in' and an 'out' hose going to the heater core. Run engine until hot, these two pipe should be both quite hot, if the out pipe feels cooler than the in pipe then it would seem the core is blocked. You say you have no leaks so maybe it would be worth removing both heater pipes, fix one side up to the water main and gently pass water through it. Hopefully lots of Brown sludge should come out.
If both pipes feel hot then the core should be ok. This means it will be an air-distribution fault and maybe the flap is stuck that changes it from hot to cold air.

pipe can be fix

it could be the freeze plugs. that can be expensive.

could be a leak or crack in a radiator hose
could be water pump not working good or leaking
could be radiator leaking
could be thermostat half blocked