Can I read from not from the beginning of a Surah?
Yes, of course you can do that, and all Imams do so, like in Taraweeh some of them read a page of Quran in every rak'ah, and then the next page.. etc so they can complete the whole Quran along the month of Ramadan.
You can start wherever you want, and read at least three tiny ayat or one non-tiny ayah. Allah said:
فَاقْرَءُوا مَا تَيَسَّرَ مِنْهُ
So recite what is easy from it [Quran]
Imam Abu Hanifah said this is an evidence that you can read the least amount of it, i.e, one ayah. However, most other scholars agree that the minimum is either 3 tiny ayat (tiny like this), or one non-tiny ayah. But this establishes that there is no restriction on the start point.
Also the Prophet (pbuh) said:
إِذَا قُمْتَ إِلَى الصَّلاَةِ فَكَبِّرْ، ثُمَّ اقْرَأْ مَا تَيَسَّرَ مَعَكَ مِنَ الْقُرْآنِ
When you stand for Prayer say Takbir and then recite from the Holy Qur'an (of what you know by heart)
Bukhari and Muslim
This also establishes that there is no restriction on where you should start.
Also, does it matter if ayahs I recite make a meaningfull context?
It's recommended that you read a contextually related ayat, especially if you're Imam, because this will encourage everence (خشوع) and ability to dive in the meanings of Quran. But it's not a must. An evidence I could say is that it's very hard to find a single non-tiny ayah that's not related to the ayat before and after it, while it's acceptable to read it alone as mentioned above. Also Allah requires us no more than we can do, and if you're non-Arabic it'll be so hard to decide if the context is ok or not.
And Allah knows best.
Source: I brought some of the info above from the book: Islamic Fiqh (Book), part 1: Prayer rules, version 8, Pages 224-227 By Dr.Ibraheem Muhammad Salqini, professor in Damascus University - College of (Islamic) Shariaa. (The book is taught in the same college).
See also What Suras/Ayat are Mustahab (recommended) to recite in the five fardh prayers.
Hope that's helpful.