Engine bucks has no power and exhaust y-pipe gets red hot.

This could be a clogged catalytic converter, or physical damage to the exhaust system. Many auto shops will check stuff like that for free, or a small diagnostic fee which they apply to the cost of the repair if you have them do the work.
You look under the car and see if any of the exhaust piping is bent or pinched blocking the flow. For the catalytic converter. After dark run the engine unit it starts to get hot then look under the car to see if any part of the exhaust is glowing orange or red. Be sure not to let the engine get too hot. If the Cat is clogged it will be glowing by the time the engine reaches normal operating temp.

which toyota engine
front exhaust manifold and pipes always get hot quick
even more quickly if have taken out the cat and bolted in cat delete pipe

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It takes a lot of heat to make the exhaust pipes get hot enough to glow red. It is a dangerous condition and can lead to a vehicle fire. A clogged catalytic converter or manifold could cause this, but ultimately the heat from the exhaust fumes is not exiting the system properly. Even a clogged tail pipe could cause this. A vandal might have stuck something in the end of the tail pipe.
It is running rough because it cannot push the exhaust out and therefore cannot take in enough air.
Regardless of the cause, have this inspected and the blockage/clog removed repaired and do not continue to drive the vehicle with this condition.

Try checking for a restriction in the system. possibly the catalytic converter. are the pipes before the muffler? you can smack your catalytic converter with a dead blow hammer a few times before you start the engine. you will hear a sound similar to breaking glass from inside the converter it's carbon build up. Start the engine and rev it up a little. If it was constricted the carbon you broke loose will blow on out. If nothing is restricted in exhaust you may have a bad exhaust valve on that side.

The exhaust system could be blocked between the engine and the tail pipe. The fuel mixture could be too rich or too lean.
Is the check engine light on ?
Can you rev the engine up in park ?

Not RED hot, no.
You may have a restriction or a timing problem.
Was it that hot before you replaced the exhaust parts ?

If the exhaust system was plugged the engine wont run, normally. It will start and idle until the back pressure is such that it can't take in more air/fuel mixture and then stall.

After the converter, there may be a resonator, and then the muffler. Is any exhaust flowing out of the tail pipe? You might try hooking up a scanner and see what your oxygen sensor(s) is/are doing.
Also, scan the MAF(mass air flow) sensor assembly.
Sound like the engine can't breathe, and its getting to much fuel.

Replace the catalitic converter. It is clooged.

on the air intake is a pipe that picks up hot air from the exhaust manifold ,where this pipe joins the air intake pipe is a vavle which is operated by a bi.metal strip thermostat ,so when its cold it allows vacumn from the engine to open the valve allowing hot air into the intake which stops it from freezing up around the butterfly ,Something is wrong around this area but without having my head under the bonnet i couldnt say ,if your in doubt ask a local mechanic to have a look they will see the problem straight away

your converter is blocked needs to be changed,unless exhaust system is damaged and causing restrition.