Jag tycker att den grå dörren ser bättre ut.

Here jag tycker is used to give an opinion. It often corresponds to I think, I feel, or I find in English.

So Jag tycker att ... is a very common way to say what you think about something.

A useful contrast:

• jag tycker = giving an opinion
• jag tror = believing something is true

For example:

• Jag tycker att den är fin = I think it is nice
• Jag tror att det kommer att regna = I think/believe it will rain

Att means that and introduces a subordinate clause.

So the structure is:

• Jag tycker = I think
• att den grå dörren ser bättre ut = that the gray door looks better

In English, that is often optional, and the same is true in Swedish in many informal contexts. So you may also hear:

• Jag tycker den grå dörren ser bättre ut

But with att, the sentence sounds very natural and clear.

Because dörr is a common gender noun, also called an en-word.

In Swedish:

• common gender nouns use den
• neuter nouns use det

So:

• en dörr = a door
• den dörren = that door / the door

If the noun were a det-word, you would use det instead.

The ending -en marks the noun as definite singular.

So:

• dörr = door
• dörren = the door

This is one of the most important patterns in Swedish: instead of always using a separate word for the, Swedish often adds a definite ending directly to the noun.

Because Swedish usually uses double definiteness when a definite noun has an adjective before it.

The pattern is:

den/det/de + adjective + noun in definite form

So:

• dörren = the door
• den grå dörren = the gray door

English only marks definiteness once, but Swedish normally marks it twice in this kind of phrase:

1. with den
2. with the definite noun form dörren

Here grå matches dörr, which is a singular en-word.

Compare:

• en grå dörr = a gray door
• ett grått hus = a gray house

So grått is used with a singular ett-word, not with dörr.

As for gråa: with some adjectives, especially color adjectives like grå, both shorter and longer forms can occur in definite/plural contexts. So den grå dörren is very natural, and den gråa dörren may also be heard. Learners will very commonly see den grå dörren.

Because se ut is the normal Swedish verb expression for look in the sense of appear.

So:

• se ut = to look / to appear

Examples:

• Hon ser trött ut = She looks tired
• Huset ser fint ut = The house looks nice

So den grå dörren ser bättre ut means the gray door looks better, not that it literally sees something.

Because se ut is a particle verb. In many Swedish sentences, the particle is separated from the finite verb.

So:

• infinitive: se ut
• present tense: ser ... ut
• past tense: såg ... ut

Examples:

• Den ser bra ut
• Han såg glad ut

In your sentence, bättre comes between the verb and the particle:

• ser bättre ut

That is completely normal.

Bättre is the comparative form of bra.

The forms are:

• bra = good / well
• bättre = better
• bäst = best

This is an irregular pattern, so you do not normally say mer bra.

So:

• ser bra ut = looks good
• ser bättre ut = looks better

No. Swedish, like English, can leave the comparison implicit if the context is already clear.

So ser bättre ut can mean:

• looks better
• looks better than the other one
• looks better than before

If you want to say the comparison explicitly, use än:

• Jag tycker att den grå dörren ser bättre ut än den vita.
• Jag tycker att den grå dörren ser bättre ut än den vita dörren.

Yes, but it means something different.

• ser bättre ut = looks better, appears better
• är bättre = is better

So:

• Den grå dörren ser bättre ut focuses on appearance
• Den grå dörren är bättre would suggest it is actually better in quality, function, etc.

That difference is very similar to English:

• looks better
• is better

Yes, very often.

Both of these are natural:

• Jag tycker att den grå dörren ser bättre ut.
• Jag tycker den grå dörren ser bättre ut.

Including att can sound a little more explicit or slightly more formal, but both versions are common and correct.