Hi there, short naps can be a sign that A times need extending, which will eventually take him to two naps. Please could you post a recent day's routine, in the form:
Up and E: 7am
S: 9-9:45 (tried to resettle without success, got up at 10:15)
E: 10:30
S: 11:30-12:30
... Or whatever it looked like for you! Please include things like how easily he fell asleep, whether you tried to resettle him, how he was on waking (eg crying or happy) and anything else you think might be relevant - eg noise waking him.
It's also worth thinking about the environment in which he's going to sleep - is it very dark? Mine were very sensitive to visual distractions and got on much better in pitch darkness, but I know others can be less sensitive. If you're having work done and it's very noisy it's also possible that may be affecting his sleep and making it more difficult for him to go back to sleep once he wakes. Do you use white noise? That can be really helpful for blocking out some noises a little.
As for the nights, I'd personally keep a feed for now, 5mo is very young to be going all night without one. Have you tried feeding at 1/2am when he first wakes? He may then manage to go through until morning without another waking. Or continue to feed at 5am if he still settles easily after that.
As for extending naps - it may be that a routine tweak will help, but otherwise how long have you been trying wake to sleep? Often people find that doing it for three days and then not doing it the fourth day (to see what happens), then doing it again for the next three days (only if needed), etc can help. Some will do it for more than three days at a stretch, but the point is it can take a little time to work. Alternatively you could wait until he wakes and give him time to resettle himself without your intervention, provided he's not really upset. He may surprise you! Or if he is upset, go in and lay a hand gently on him, say your sleepy phrase if you have one, or just shh if he's used to that, and walk back out again, as if at the start of the nap. Wait a while to see what happens, if he's crying, go back in and repeat... Or, at this age, some LOs who can settle themselves at the start of a nap need more help to go back off half way through, so you might have more success staying in the room for longer, or laying a hand on him until he's almost asleep. Experiment to see what seems to help him most - don't forget, you know him best