Mein Rasierer liegt im Bad neben dem Waschbecken.

Because Rasierer is a masculine noun and it is in the nominative case here, since it is the subject of the sentence.

For a masculine noun in the nominative, the possessive determiner takes the form mein:

• mein Rasierer = my razor

Compare:

• der Rasierer → nominative masculine
• mein Rasierer → my razor

Meiner by itself can sometimes mean mine, but it does not fit before the noun here.

Rasierer is masculine.

Its dictionary form is:

• der Rasierer = razor / shaver

That is why you see masculine forms around it, such as mein in the nominative.

German often uses different verbs for where something is located, depending on its position.

liegen is commonly used for things that are:

• lying down
• resting flat
• simply located somewhere, especially smaller objects

So Mein Rasierer liegt ... is very natural for a razor.

Compare:

• Das Buch liegt auf dem Tisch. = The book is lying on the table.
• Die Flasche steht auf dem Tisch. = The bottle is standing on the table.

German often prefers liegen/stehen/sitzen where English would often just say is.

Yes, Mein Rasierer ist im Bad ... would be understandable, but liegt sounds more natural and specific in German.

Using ist only tells us that the razor exists or is located there.
Using liegt gives a more natural sense of physical position.

So:

• ist = possible, but less idiomatic here
• liegt = the better, more natural choice

Im is simply the contraction of in dem:

• in dem → im

So:

• im Bad = in dem Bad

German very often uses this contraction, and it is the normal form in everyday language.

Good question. Bad is indeed neuter:

• das Bad

But after the preposition in when it describes location rather than movement, German uses the dative case.

The dative form of das is dem:

• nominative: das Bad
• dative: dem Bad

So:

• in dem Bad / im Bad = in the bathroom

Because neben is a two-way preposition. That means it can take:

• dative for location
• accusative for movement toward a place

Here the razor is already located next to the sink, so German uses the dative:

• neben dem Waschbecken = next to the sink

If there were movement, you would use the accusative:

• Ich lege den Rasierer neben das Waschbecken.
  = I put the razor next to the sink.

So:

• location → neben dem
• movement → neben das

It is dative singular.

The noun is:

• das Waschbecken = the sink

Because neben is describing where the razor is, not motion, it takes the dative here:

• neben dem Waschbecken

So the article changes like this:

• nominative: das Waschbecken
• dative: dem Waschbecken

Das Waschbecken usually means sink or washbasin.

It is made up of:

• waschen = to wash
• Becken = basin

In a bathroom sentence like this, sink is the most natural English meaning.

German main clauses normally put the finite verb in second position.

So the structure is:

• Mein Rasierer = subject
• liegt = finite verb
• im Bad neben dem Waschbecken = information about location

That gives:

• Mein Rasierer liegt im Bad neben dem Waschbecken.

The location phrases come after the verb. This is very normal German sentence order.

Yes, but in many contexts they overlap.

• das Bad often means bathroom
• das Badezimmer also means bathroom, and is a bit more explicit

In everyday German, im Bad is extremely common and natural.

So in this sentence, im Bad sounds perfectly normal.

Because in German, all nouns are capitalized.

So:

• der Rasierer
• das Bad
• das Waschbecken

This is one of the most noticeable spelling rules in German.

Yes, it can, and the sentence would still be grammatical:

• Mein Rasierer liegt neben dem Waschbecken im Bad.

Both versions are possible. The difference is mostly one of focus and style.

• im Bad neben dem Waschbecken first places the razor in the bathroom, then gives a more precise location
• neben dem Waschbecken im Bad first gives the immediate location next to the sink

The original sentence sounds very natural.