the engines a V so it should have two one for each side for each camshaft.

the chip in the key is a transponder and there is also one in or around the barrel or behind the dash and the frequencies marry up to each other in order to start the car and does hand in hand work with the immobiliser so the transponder in the car may be a possibility don’t rule it out fully. could be faulty wiring or water getting into the system. if that green key light is on and off intermittently my guess would be loose wire or water shorting something in the system every so often which is immobilizing the car and that’s why its shut off whilst driving but not all the time

check the coolant temp sensors and see if there faulty first, check your fuses and make sure there all okay and that the wiring harness that is plugged into the sensors and the fan module behind the radiator are all okay and no signs of anything loose or broken. when you drive the car is the coolant temp within normal range?

could be a number of things, any particular driving cause it to happen or anything which may help pinpoint it i.e does it get up to temp, any odd noises or does it stop at a certain point of driving? First thing I would do check all serviceable parts like filters and include the fuel filters, plugs and air system to rule out any blockages. If it hasn’t been serviced recently then get it done, may help the issue. Most common things would be air, fuel, compression in the combustion chamber is having an issue. Other possibilities could be a vacuum leak (if its turbocharged), faulty crank position sensor, faulty coolant sensor, alternator issues or at a long shot if the timing if slightly off it can cause an engine to stall but usually there’s other tell tale factors with that but depending on mileage could be worth a check if all other possibilities haven been looked at. if its any sensor for the engine like the crank sensor or any others related to the engine itself they will usually throw up a CEL light and...

Check fuses and relays first as jumping it should bypass the battery even if its shot and start the car or crank it over, there’s two fuse boxes one in the engine bay and one on the passenger side of the dash, if you open the door there will be a small handle to pry the plastic trim away and its behind there. Another thing to consider is if the battery was discharged to a low amount of power usually if it drops below 10v it can damage them and they won’t hold charge very well even if you re-charge them again or try every trick in the book so get the battery tested as this may be making the problem more complicated than it is now or may well continue to do so, there’s usually a date stamped on the battery to indicate when it will most likely need a service or recharge in a sense) Normally if the battery is damaged one of the signs is slight corrosion on the terminals. If you battery is healthy and the fuses and relays are all okay then If you want to continue to try and fix the issue yourself I would also...

if there’s lights on the dash indicating an issue then get a diagnostic tool plugged in, lots of BMW’s once these lights are on won’t allow you to start the car for safety reasons probably to protect the engine. BMW specialist or a in depth diagnostic tool to read it before taking any guesses rather than a standard OBD tool, The light comes on for a reason, clearing it won’t do anything other than just come back on and cause more damage more than likely continuing to drive it. the icon with the car on a lift is an indication built in to the car meaning it is due for a service, if it hasn’t been done recently may explain why the coolant light is on too.

let the motor get cold, drain the system, refill it with the correct coolant/water mix to the max level, bleed the system (most important thing to do or this will make it overheat with no obvious culprits). Turn the engine on and let it idle and get up to temp with the hood open and just watch for any leaks underneath too, even a small one may take a while to notice, keep and eye on the temp though and turn off if you need to. If you can’t find a leak then that’s narrowed it down a bit. if you’ve already replaced the water pump and thermostat and are confident that they are functioning as they should I would check the radiator and fan next.

does it start up okay without any issues? if your getting very weak acceleration and starting with a bit of a stumble every now and then on idle I’d check the plugs, fuel system and filter which you’ve already done and air intake first of all. could check for any clogged oil lines, I know you’ve ran sea foam but if there was already a pretty bad blockage then using it to loosen up other deposits may have collected in the same place. compression could also be a tad low, is your oil consumption normal?

any small replacements on the dash check Ebay, Amazon etc… failing that go to the dealer or ask your regular mechanic or a supplier.

if its an auto then downshifting is probably what’s happening, most gearboxes have that difference in ratio. depending on where you are driving it could be normal for it to downshift like if your climbing up a gradient and the engine is struggling or if you are putting the accelerator more than around half way down the ecu will be programmed to downshift. If it is an auto and you have the option for a manual mode try using that and go out for a drive, if you select one gear that’s comfortable for 70mph and go on for a mile or so as long as you have it in the manual mode it shouldn’t change. If this fixes the problem for you then you have narrowed you issue down and can get someone who knows a bit about transmissions to take a look.