Youâve got a couple of things happening there.
SMS/text messages go via the modem, not wifi, and they stay on your ISPâs server for a while until the device which they are intended can and has received them (for how long they may wait there for that to happen depends on service but my experience has been about upto three days before being trashed - can be less). So, those will arrive, when the modem is awake and connected to network.
If the phone is set to go to suspend mode (from Settings â Power), Wifi goes to suspend mode too and stops/disconnects to save power. There are steps to saving power (which you can select at setitngs), starting with screen dimming after a set period of inactivity (screen takes a lot of power) and screen shutting down (goes to black), but those do not affect the modem. So called âsleep modeâ isnât a term that is used but from phone behavior, the suspension would be closer to it than just energy saving from screen dim and blanking. What suspension also does, is it also shuts down CPU (so nothing is doing anything, even in the background), except (if I remember this right and can simplify it) there is are tiny periodic momentary wakeups to check if modem (phone) or something else needs to come back online (like get SMSs).
This is to say, what you may see from the icons may not be the card shutting down but rather fluctuations in connectivity (as in, wifi not having stable signal to router, maybe die to interference or being moved etc.) or something else than shutting down (powering off, going to suspension, going to sleep). That being said, wifi should also try to go to some lesser state of activity if there is no data to transfer, just to same power - itâs not supposed to keep constant data flow if there is no need for it - and come right back when itâs needed again (which may cause a slight reactivation lag, which is normal). Bluetooth, since itâs on the same card, pretty much behaves according to to the same power saving rules, except itâs more efficient. I donât think there are setting available (or even possible) to have wifi or BT activate and wake the phone up when they are set to suspend.
From the forum you can find some advice where you can tweak some of these parameters for power, suspend, wifi, modem etc. (if really needed). But, depending on your settings, that power button push either wakes up the screen (if itâs dimmed/blanked) and doesnât effect the modem or wifi, or it wakes the phone from suspend - check your setting what you have selected it to do (suspend can be selected separately for when being only on battery and when itâs plugged to a power source).
(not the top slide, as in the screen grab), where in the list there is a Power tab, where there is a setting for switching suspend (Automatic Suspend: on battery, and when plugged in, and their delays). That will determine if the suspension activates after a period of inactivity, taking the device to âdeeper sleepâ (from where it needs to be woken up with power button press, or by a call). After waking up, everything should resume as it was - connections and all (suspend does not set software switches to off). Unfortunately I canât show you a screengrab of that settings tab as I have the backports version, which looks a bit different than normal byzantium.
Those switches in 
will do.