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Invoke a custom code on internet connectivity in Android

I want to get latest updates from my server when I turn on the internet and generate a notification, but when the app is closed/killed/swiped from recent items, there is no way to keep my service alive and listening to network change event so that I can ping my server, I'm not sure how to do it and how other apps such as whatsapp does it when we receive new notification the moment we turn on mobile internet.
Any help is appreciated, thanks in advance.

Making clear what #GobuCSG commented. You have the following options:
Run your application all the time via a foreground service and listen for wifi connected broadcast. This is not a good solution and wastes battery life.
Schedule jobs using the WorkManager API.
The WorkManager API was developed specifically for the purpose you specified. The API allows you to set required conditions for your job to run, such as network connectivity. It also persists scheduled jobs through device reboots, so you don't have to. The only downside is that you don't have fine tune control of when the code runs, as one of the goals of WorkManager is to save battery life by delaying and batching jobs.
EDIT:
Another option is to use push notifications. This is useful if your are developing a messaging app and you want your server to push a message to the client so they can be notified of a received message. But, if all you need to do is establish contact with your server once a day, then you should use WorkManager.

Android - reliably ping server every few seconds

I have a business requirement that the Android app needs to report to the server every few seconds (all the time, 24/7) on a dedicated device.
First I thought that it can be done with the PeriodicWorkRequest, but I've read its minimum interval is 15mins.
What would be the best way to achieve this? What mechanism I can rely on to be sure that the process won't be killed?
Is it possible to do with WorkManager? Should I have a foreground service with a loop?

I think you could schedule it the other way. Make your server send a notification to your android device,could be done easily via firebase. Then handle notification in your app and report back to the server.

Schedule task when other app is using network

In my app I show an persistent notification on status bar in which real time data is shown. However I cannot just make a poller for the real time data because it will cost too much battery. What I am thinking is that I update the data whenever some other app is using the network. So that network requests are batched and battery consumption will be reduced. Is there an api in Android to do such scheduling? Thanks!

Yes, you can achieve scheduling tasks by using the AlarmManager class, you can have look at the offical doc.

Syncing the website account with android app

i was thinking if there is anyway to sync android app with a web/website account. Basically, if the user logs in through the web, i want the android app to log in/start up/show notification. Should i use a background service in android to monitor the account or something else
Please help
Thanks

As far as I can see, you have a couple of options:
Use a background service (as you suggested) which is constantly running and checking. This will be battery consuming.
Use the AlarmManager to schedule a repeating alarm to check on a scheduled interval. This will be less battery consuming but also less immediate (unless you set a low interval which will consume a lot of battery).
Use a push message. This will be the most accurate and least battery consuming for your phone, but it's also the most advanced (difficult) to set up.

Based on your requirement, optimized solution is to use GCM - Push Notification. Based on the whatever action occurs at web, you just need to send a push notification to clients. Based on the push notification IDs, client needs to perform appropriate action.

Best way to have an Android app poll periodically in the background

Take the Gmail app as an example. Whether the phone is on or not, it polls every 10 minutes or so to download new emails which may have arrived since you last checked.
I know how to create a new service and bind to it. But I can see a few ways to accomplish this:
Bind once, and have the service run in an infinite loop, sleeping for 10 minutes between each loop
Bind and unbind right when it's done, scheduling the next bind somehow in 10 minutes
Using the AlarmManager class to schedule future polls
What are the trade offs? How does the Gmail app accomplish it?
Thanks!

Gmail app uses pushing, not polling. I suggest using this technique instead, polling is a battery killer in mobile devices.
To implement pushing, take a look at C2DM.
If you still want to poll, the recommended way would be to set up a periodic alarm in the AlarmManager.
UPDATE: Google has deprecated C2DM and replaced it with Google Cloud Messaging (GCM)
UPDATE: Google has deprecated GCM and replaced it with
Firebase Cloud Messaging (FCM)

For a continuous, but not intensive poll like the one you comment (in the range of minutes between polls), I would implement it with AlarmManager. That way you make sure the phone wakes up to poll without the need for a wakelock, which would destroy your battery. As CommonsWare pointed out, you will still need to implement a wakelock for the time your code is executing, but you can release it as soon as the code is done, avoiding keeping the phone on while just waiting. See his comment for an example on how to implement it.
I would use a Service if, instead, you need faster polls during a shorter period of time (seconds between each poll), since setting alarms does not make sense to such short periods, and the battery would drain anyway.