Šiandieniniaos Prancūzų patarlė yra „mažai upeliai sudaro dideles upes“.

Šiandieninė could also probably work, but šiandienos sounds better to my ear. Šiandienos patarlė (whose? = today's) vs Šiandieninė patarlė (what kind? = today's)

Tai reiškia, kad dideli pasiekimai pagamintairasideda nuo mažių žingsnių ir mažių poelgių.

I wasn't too sure how to correct the main verb here. This says that big accomplishments begin with small actions (which eventually turn into the "big river"). Not sure if that takes away from the intended meaning :/
"kyla iš mažų poelgių" could work too (rises from). However, that requires removing the steps part (the verb "kyla" is simply incompatible here")
Ack, translating proverbs is not easy!

AKalbant apie upes...

Seine upės šaltinis yra netoli iš mano namoų.

Netoli does not go with "iš". Depending on the situation, it can go with the genitive case, or sometimes with "nuo" + genitive case. Unfortunately, I don't know the logic behind when it can go with nuo :'D All examples I can think of sound good without "nuo", so I think it's best to stick with that!

Jau tenTen esu vairavauęs.

We use būti + "adjectival participle" form (that's what Dalyvis is called, according to google :D) to describe present perfect.
I would remove "jau"/"already" completely. It makes me expect that you will compare it to other places where you haven't driven before, because you are mentioning such detail.

Sequanos statula (keltų dievė, iš kurios kilo upės vardas kalęs) dar jį saugoja.

Kai mažią sroves mačiaučiau tik mažą srovę, man bauvo keista galvoti, kad ji taps šia didele upe, kuri teka per Paryžių teka.

About the word order -- Lithuanian is "easy" (debatable xD) in a sense that you can put the words in whatever order, since we can understand what role in the sentence it acts based on its ending. However, the "standard"/most neutral sentence structure is Subject-Verb-Object (S-V-O).
If you put the Object before the verb (S-O-V), it has the consequence of putting subtle emphasis on it (the Object). Same thing with O-V-S -- puts emphasis on the Subject. That can be somewhat prevented when speaking, since intonation has the final say where that emphasis actually is, but in written form, it is much more important.

In a simpler sentence:
- Mano sesė sodina gėles. = My sister plants flowers. (regular sentence, no emphasis unless one is indicated when speaking)
- Gėles sodina mano sesė. = *My sister* plants flowers. (It's not me, and it's not my brother, it's my sister specifically that plants flowers).
- Mano sesė gėles sodina. = My sister plants *flowers*. (She's does not plant tomatoes or carrots, she specifically plants flowers).

Fun fact: There is a phrase that breaks this standard word order -- "I love you"! We say "Aš tave myliu" (S-O-V), and it doesn't feel like any extra emphasis is put on the word "you"! "Aš myliu tave" is also acceptable, of course, but in my subjective experience, it only happens in songs (usually for better rhyming), or when you want to put emphasis/contrast on "you" (as in, "I love *you*, not somebody else).

Gamta visada yra nuostabos šaltinis!

It is completely understandable, although to my ear, it sounds a bit like a literally translated sentence. That could be just my subjective opinion though :')
I think saying something like "Gamta nenustoja stebinti" (Nature never stops amazing (me/us/everyone)) sounds better