A bit about the relevant fiqh rulings and fiqh views
First you should know that the sujud or prostration of the recitation (reading of the qur'an) also know as sujud at-Tilawa سجود التلاوة is not an obligatory act, you may do it or leave it as it is optional. An evidence for that can be found in Muwatta' Malik here.
Secondly there's no 100% consensus by now when it comes to whether sujud of thankfulness or sujud of recitation are considered as equal to sujud while praying. Those who consider them equal to the sujud of the prayer or even a prayer themselves require the same requirements for these kinds of sujud to those of the prayer namely: taharah (purity) and the hiding of 'awrah ( for example wearing hijab for women). And they might have strong evidences for that based on the assumption that this sujud is a prayer by itself, imam a-Shafi'i even gave it a related term.
Nevertheless most scholars and those of the four sunni madhhabs hold the opinion that sujud for recitation of the qur'an requires taharah (purity at least by wudu' or if necessary ghusl). There's no strong backup for this as we don't know whether the prophet () himself considered equal to a prayer, but this is a strong assumption which found a big consensus among the scholar, while among the Sahaba ibn 'Omar and later scholars like a-Sha'abi, imam al-Bukhari, ibn Hazm, ibn Taymiyyah, a-Shawkani and most modern salafi's actually disagree with this opinion and used some evidences to refute it: For example the sahih hadith (See for example here in Sahih al-Bukhari "Chapter: The prostration of Muslims along with Al-Mushrikun; and a Mushrik is Najasun (impure) and does not perform ablution" See also the commentary of ibn Hajar here in Arabic) about the prostration of the Muslims and disbelievers after hearing the verses from surat an-Najm can be used as an evidence refuting this view. As none can certify that all Muslims were in a pure state nor do the disbelievers!
Also the sujud of recitation of the qur'an and that of thankfulness is not a prayer so both shouldn't basically require the same conditions and requirements of a prayer as long as there's no proof by the practice of the prophet () showing this as none of the sahaba reported that all of the congregation had wudu' when they prostrated with the prophet (). And ibn Qudamah even reported that 'Othman ibn 'Affan considered a likewise prostration as permissible for a woman in her menses.
Nevertheless it is better to recite the qur'an in a pure state and it is certainly better for a believing woman to wear her hijab while reciting. But as said above there's no clear evidence for neither so you could prostrate without wudu' or taharah (for example if you were listening to a qur'an recitation while in your menses or reciting from your own memory) and you don't need to wear a hijab while reciting. Ibn Qudamah (Hanbali scholar) even said there's consensus among scholars about the fact that prostration (no matter which) requires all that which is required for a prayer, but as stated above the evidence for this opinion is somewhat weak and a consensus as stated doesn't exist as I've already quoted several disagreeing opinions.
Note that for the prostration while praying both are necessary conditions for the validity of a woman's prayer no matter what kind of prayer she is performing.
Be aware that if you hold a Moshaf in your hands and you passed by a verse of prostration you should be tahir as the rulings stated above in first place hold for the case of a recitation out of your memory or one you were listening to.
Finally note that reading the qur'an with your eyes only is inadequate you should read it with your lips as this is the only way to get rewards for your recitation.
How to
When you reach a verse of sujud and end it you should turn to the qiblah (if you were not facing it before) and say: "Allahu akbar" and perform a normal sujud there are actually a few du'a's which one could chose to say, but saying the usual ones which one would say in a prostration of the daily prayers would be ok (See for example this one in Sunan an-Nasa-i)! There's also a dispute among scholars whether this sujud requires a raising of hands at the moment of takbir (when one says "Allahu akbar") in the case of a takbir of recitation outside or during a prayer. Once you finished your prostration you go ahead with your recitation (you may say "Allahu akbar" when finishing your prostration -as per the 4 sunni schools of jurisprudence- or leave it, unless you've performed it while praying or "As-Salamu 'alaikum" or anything to end it, as there's no evidence for that).
As for putting the qur'an aside. Note that it is worth saying that you can't put the qur'an aside, but you would put a moshaf aside. As the qur'an is Allah's word which was orally transmitted while the book copy you may read from is a Moshaf and not the qur'an itself. Also note we are not required to worship the qur'an nor a Moshaf, but respect it by not putting it on a non-tahir surface. The respect of the Moshafs may differ from madhhab to madhhab and from person to person. But what I've mentioned is just the lowest level of respect.
Signs in Moshaf
Note that earlier scholars would consider any sign in the Moshaf as bid'ah (as the earliest Moshafs didn't have any additional signs or letters such as verse numbers etc.) and I wonder if there's any consensus about a specific sign mostly you would find at least a side note for a "sajdah" "سجدة" and some kind of symbol like this:
which is one of the symbols the Arabic wikipedia offers. Note that in the Moshaf of Medina which is printed in Saudi Arabia and likewise the above or a similar symbol is used and usually the "Reason" for performing sujud is either under -or over- lined prior to putting this symbol at the end of the relevant verse. So once you see this line you should go ahead reciting and not stop even if in cases it may occur that it appears one verse earlier.
References:
islamqa #4862, #22650, #220213 (in Arabic or Urdu) and #235969 (in Arabic only)
and in al-Mawso'a al-Fiqhiya الموسوعة الفقهية
and on islamweb #2769
As for your question about other symbols in the qur'an I don't really get what you mean.
Also relevant:
Can I perform sujud shukr without wudu/ablution?
Is sujood permissible outside salat when it is only for asking Dua?
Number of Prostrations in Quran in every Juz (1/30th part of Quran)?
