Striken og keglefaldet står på scoretavlen.

Because Danish usually puts the definite article the at the end of the noun.

• en strike → striken = the strike
• et keglefald → keglefaldet = the pinfall
• en scoretavle → scoretavlen = the scoreboard

So instead of a separate word like English the, Danish often uses a suffix.

That depends on the noun’s grammatical gender.

• en strike is a common-gender noun, so the definite form ends in -en
• en scoretavle is also common gender, so it also takes -en
• et keglefald is a neuter noun, so the definite form ends in -et

This is a very common pattern in Danish.

It is a compound noun:

• kegle = pin
• fald = fall

So keglefald literally means something like pin-fall, and in bowling it refers to the number of pins knocked down. As in many Danish compounds, the last part determines the gender, so because fald is neuter, keglefald is also neuter: et keglefald.

Because Danish often uses stå for something that is written, shown, or displayed on a surface.

So står på scoretavlen does not literally mean that something is physically standing there. It means it appears on or is shown on the scoreboard.

You will see the same idea in sentences like:

• Det står på tavlen. = It is written on the board.
• Hans navn står på listen. = His name is on the list.

Using er is sometimes possible, but står på is more idiomatic here.

Because Danish verbs do not change form for singular and plural in the present tense.

So:

• Han står = he stands / is standing
• De står = they stand / are standing
• Striken og keglefaldet står = the strike and the pinfall are shown

Unlike English, Danish does not have a special plural present-tense form here.

Because a scoreboard is treated as a surface or display, and Danish normally uses på for things that are on boards, lists, screens, tables, and similar surfaces.

So:

• på tavlen = on the board
• på listen = on the list
• på skærmen = on the screen
• på scoretavlen = on the scoreboard

Using i would sound wrong here because the information is not thought of as being inside the scoreboard.

Yes. Danish very often makes compounds, and it also freely uses loanwords in them.

So scoretavle is a normal compound:

• score
• tavle = board

Likewise, strike is an English loanword used in Danish bowling language, and it behaves like a Danish noun:

• en strike
• striken

Loanwords often take normal Danish endings once they are used as part of everyday vocabulary.

Yes, absolutely. That is also correct Danish.

Danish is a verb-second language, which means the finite verb usually comes in the second position in main clauses. So if you move På scoretavlen to the front, the verb still stays second:

• Striken og keglefaldet står på scoretavlen.
• På scoretavlen står striken og keglefaldet.

The meaning is basically the same, but the second version puts more focus on på scoretavlen.