Odpadní hadice povede mezi skříní a stěnou.

Because odpadní is a soft adjective. Soft adjectives have the same nominative singular form for all genders:

• jarní den = spring day
• jarní voda = spring water
• jarní počasí = spring weather

So even though hadice is feminine, the correct form is still odpadní.

It is nominative singular, because it is the subject of the sentence.

You can tell because:

• the verb povede is 3rd person singular
• hadice is the thing that will run / will be routed

So the basic structure is:

Odpadní hadice + povede + mezi skříní a stěnou
= The drain hose + will run + between the cabinet and the wall

Here povede is the future tense, 3rd person singular, of vést.

The verb vést can mean:

• to lead
• to run
• to go along
• to be routed

With things like hoses, cables, pipes, roads, and wires, Czech often uses vést where English uses run.

So povede here means something like will run or will be routed.

Also, this is not the same as povést se meaning to turn out well / to succeed.

Yes. This is very natural in Czech.

Czech uses vést for the route or course of many things:

• Kabel vede pod podlahou. = The cable runs under the floor.
• Trubka vede do sklepa. = The pipe runs into the cellar.
• Cesta vede lesem. = The path goes through the forest.

So Odpadní hadice povede... is a normal way to say that the hose will run somewhere.

Because mezi can take different cases, and here it takes the instrumental.

In this sentence, the phrase describes where the hose will be routed / located:
between the cabinet and the wall

So Czech uses the instrumental:

• mezi skříní
• a stěnou

A useful contrast is:

• Hadice povede mezi skříní a stěnou.
  The hose will run between the cabinet and the wall.
  → location / routing

• Dám hadici mezi skříň a stěnu.
  I will put the hose between the cabinet and the wall.
  → movement into that space

Because skříň and stěna belong to different noun patterns.

• skříň is a feminine noun ending in a soft consonant
  → instrumental singular: skříní

• stěna is a feminine noun ending in -a
  → instrumental singular: stěnou

So both words are in the same case, but they take different endings because they decline differently.

It is singular here.

The base word is skříň = cabinet / cupboard / wardrobe.
Its instrumental singular form is skříní.

That can be confusing, because skříní also looks like some other Czech forms, but in this sentence the case is clear because it follows mezi and is paired with stěnou.

Because Czech does not have articles like a and the.

So odpadní hadice can mean:

• a drain hose
• the drain hose

Which one is best in English depends on context. In instructions, English often prefers the drain hose, but Czech simply says odpadní hadice.

No. Czech word order is fairly flexible.

Odpadní hadice povede mezi skříní a stěnou. is a neutral way to say it.

You could also say:

Mezi skříní a stěnou povede odpadní hadice.

That still means the same thing, but it gives more emphasis to where the hose will run.

So the original sentence is not the only possible order, just the most straightforward one.