Fellow health anxiety sufferer here, and mum to two children who have caused me many sleepless nights over last few years. A few things….
38.4 is high, but it’s not alarmingly high - and will absolutely not worry a doctor if a child is otherwise eating, drinking, weeing and ok in themselves (responsive, interested in their adult, talking, smiling).
The cough - again, my son get terrible nigh time coughs when he has a cold (their tubes are very small when they’re little and some are disproportionately impacted and cough more than others). If it’s a cough where he struggles to recover, or sounds wheezy in between, then he may need some help (usually his is a viral wheeze and they need an inhaler and spacer and maybe a quick whiff of oxygen). If it’s mostly at night and sounds mucusy, it’s probably nasal drip (still awful - used to make my son vomit). The best intervention - we tried many - is a humidifier - absolute game changer.
I’d call 111 if you’re worried. They will only really worry if there is a sign of bacterial infection (puss on tonsils etc), signs of infection that’s overwhelming (not weeing, very ‘off’, mottled skin, very cold hands and feet), or if his oxygen saturations are low (wheezy, cough through day and night, feeling off).
Lastly, I manage my anxiety by being able to do my own observations at home - temperature, oxygen saturdations, heart rate, respiratory rate and capillary refills. They don’t stop me taking him if I’m really worried obviously, but they do calm me when everything else is ok other than a mild temp and a tickly cough in an otherwise happy but snotty child.
Sorry ridiculously long post - I should say all of this is framed in my son having had sepsis at 1 and being admitted for a viral wheeze (oxygen level dipped). So I’ve seen him when he’s very sick vs when he’s got a virus / cold.