Het bed voelt stevig aan.

Because bed is a het-word in Dutch. Dutch nouns take either de or het, and bed happens to be neuter, so it uses het.

So:

• het bed = the bed
• een bed = a bed

This is something you usually have to learn together with the noun.

Because the subject is het bed, which is third-person singular.

The verb is voelen, and in the present tense:

• ik voel
• jij/u voelt
• hij/zij/het voelt
• wij/jullie/zij voelen

Since het bed is singular, Dutch uses voelt.

Here aan is not really a separate word with its own independent meaning like on or at. It is part of the separable verb aanvoelen.

• aanvoelen = to feel, to seem by touch, to come across as

So:

• Het bed voelt stevig aan = literally something like The bed feels firm (to the touch)

In the main clause, separable verbs split:

• aanvoelen → voelt ... aan

That is why aan appears at the end.

Because aanvoelen is a separable verb.

In a normal main clause, Dutch separates the verb:

• Het bed voelt stevig aan.

But in places where the verb stays together, it becomes:

• omdat het bed stevig aanvoelt
• het stevige aanvoelen van het bed
• Het bed kan stevig aanvoelen

So the ending aan is not random word order; it is the separated prefix of aanvoelen.

Because stevig is used here as a predicate adjective, not directly before a noun.

Compare:

• het stevige bed = the firm bed
  Here the adjective comes before the noun, so it usually takes -e.

• Het bed voelt stevig aan.
  Here the adjective comes after the verb and describes the subject, so it stays in the basic form: stevig.

This is very similar to English:

• a firm bed
• the bed feels firm

They are close, but not exactly the same.

• Het bed voelt stevig aan focuses on the impression you get from touching, lying on, or testing the bed.
• Het bed is stevig is a more direct statement that the bed is firm.

So voelt ... aan is often a little more subjective or sensory:

• how it seems
• how it comes across
• how it feels in practice

English makes a similar distinction:

• The bed feels firm
• The bed is firm

Sometimes learners may hear or imagine that, but stevig aanvoelen / voelt stevig aan is the more natural and standard phrasing for this kind of physical impression.

Using aanvoelen is very common with adjectives such as:

• zacht = soft
• hard = hard
• stevig = firm
• rauw = rough
• koud = cold

So Het bed voelt stevig aan is the safest and most idiomatic version.

Good question. In this sentence, voelen does not mean that the bed itself has feelings. It works like English to feel in sentences such as:

• The pillow feels soft.
• This chair feels uncomfortable.

So the subject is the thing being experienced, and the meaning is:

• it feels a certain way to us

Dutch uses this construction very naturally too:

• De stof voelt zacht aan.
• De muur voelt koud aan.
• Het bed voelt stevig aan.

Yes, it is present tense: voelt.

Here it expresses a current general quality of the bed:

• when you try it now, it feels firm
• in general, it has that quality

Dutch uses the simple present for this very often, just like English:

• The bed feels firm.
• Het bed voelt stevig aan.

In a subordinate clause, the separable verb usually comes back together as aanvoelt.

For example:

• Ik denk dat het bed stevig aanvoelt.
• Omdat het bed stevig aanvoelt, wil ik het kopen.

So compare:

• Main clause: Het bed voelt stevig aan.
• Subordinate clause: ... dat het bed stevig aanvoelt.

This is an important pattern with separable verbs in Dutch.

Stevig often means firm, solid, or sturdy, depending on context.

For example:

• een stevig bed = a firm/sturdy bed
• een stevige tafel = a sturdy table
• een stevige wind = a strong wind
• stevige schoenen = solid/sturdy shoes

So in this sentence, stevig most naturally means firm because we are talking about how the bed feels.