How to know if someone downloaded an app to my iPhone and then deleted it?

The app will appear in your purchased apps.

View your purchase history for the App Store, iTunes Store, and other Apple media services - Apple Support

Note that it will only appear once. You won't be able to tell if it was downloaded and deleted repeatedly.

As noted above, put a passcode on your phone and this will stop.

If other people have access to your iPhone or other Apple device, change the device passcode and never share it with others.

Change your Passcode -> Use a passcode with your iPhone, iPad, or iPod touch - Apple Support

Change your Apple ID password -> Change your Apple ID password - Apple Support

• Create a strong password for your Apple ID, It is best if your password consists of eight or more characters and include upper and lowercase letters, and at least one number. You can also add extra characters and punctuation marks to make your password even stronger. 
• Never provide your password, passcode, security questions, verification codes, recovery key or any other account security details to anyone else. Apple will never ask you for this information.

Then see this Apple article: What to do after you change your Apple ID or password

What to do after you change your Apple ID or password - Apple Support

(if you haven't already done this) Two-factor authentication for Apple ID -> Two-factor authentication for Apple ID - Apple Support

To keep your Apple ID account secure, see the following Apple articles: 

If you think your Apple ID has been compromised

If you think your Apple ID has been compromised - Apple Support

See who has access to your device or accounts -> See who has access to your iPhone or iPad – Apple Support (UK)

Delete suspicious content from your devices -> Delete suspicious content from your devices – Apple Support (UK)

Check your Apple ID device list to see where you're signed in

Check your Apple ID device list to find where you're signed in - Apple Support

• if you see devices that don't belong to you, delete from your account.

Personal Safety User Guide -> Personal Safety User Guide – Apple Support (UK)

Manage and use your Apple ID -> Manage and use your Apple ID – Apple Support (UK)

Never jailbroken your iPhone.

No

If I thought someone might be untrustworthy, you can bet I wouldn't let them have the passcode to my phone. But, you do what works for you.