I've only done short day hikes along different sections but perhaps I can help a bit with some general info.
I am a solo traveler, is the trail busy this time of the year (I hope I will find some hiker on route).
September and October are considered two of the best months for hiking in the area as July and August are extremely hot and humid.
Is the weather/water still warm?
Some mornings may be rather cool but in general the weather is glorious with very little chance of rain until later in October. It would be very unusual for the sea to be considered "too cool".
Unfortunately my time does not allow me to go all the way therefore I will need to skip some sections by taking bus. Do you have any recommendation which section is less interesting? I enjoy the beaches but also climbing mountains so hard to choose ( I would skip a section like the Meseta on the camino as an example).
In general the Lycian coast is very rugged with mountains rising directly from the sea. This is why very little of the trail is in view of the sea. It also means, unfortunately, that the trail is on the other side of mountains from what is quite arguably the most beautiful stretch of coast in the country (it is between Kalkan and Kaş). The few sections of coast that are low in elevation and at least reasonably flat find highly intensive agriculture--frequently in the form of "seas" of greenhouses filled with bananas, tomatoes, citrus, etc. In these areas there is usually reed-filled brackish band of varying width at the coast itself with the greenhouses starting as soon as the land is high enough that salt water does not intrude. While the definition of "mountain" varies, you will have PLENTY of climbs and descents!!!! Also long sections where you are lucky to even glimpse the sea. Sorry, but I don't know enough about the trail to suggest sections to skip.
What sort of average cost I should calculate (I am not a big spender..)if let's say I will stay in pensiyon every other day and camp on the next? Is it better to take euros/pounds and change money there?
While I may be underestimating I believe 10 pounds/day for food and 10 pounds/night for a simple pension are possible. I would however at least double that price for accommodation if you are within a mile of a true beach. Presuming you're from the U.K. take pounds. While there won't be many places to change money along the hike you can change money in Fethiye. Pounds are very widely accepted so in a pinch you're unlikely to have any problem using them. Efes (the major Turkish brand of beer) is a decent pilsner with the lowest price (draft especially) of any type of alcohol. Rakı, some Turkish wines and local vodka/rum are also reasonably priced. Anything imported is quite expensive.
I am preparing my camping gear based on my trip to the camino in Spain, is there anything on this route that you could not do without or something that you took but regretted later on?
Two things I suggest: insect repellent (mosquitoes are VERY dense in some areas) and water purification tablets (if you need to get water from an "iffy" source). On the subject of mosquitoes: Particularly if you're one of those people who seem to attract them I suggest getting some ointment(s) from a pharmacy in Fethiye before you begin. Turkish mosquitoes burn and itch INTENSELY!!!!
is there wifi on the bars/restaurants generally?
WiFi is nearly universal in places of accommodation at all levels. That said, I know that parts of the Lycian way are quite remote and pass through only the tiniest of villages. Wi-Fi in restaurants--particularly the simple inexpensive ones--isn't as common. I've never had a need for it at any bar in Turkey as I'm one of those crazy people who refuses to own a smart phone or tablet.
anything to mention regarding safety?
Nothing that does not apply to general travel in Western Europe. Despite news you hear, Turkey is an extremely safe country where violent crime is rare and even property crimes of opportunity are less common than nearly anyplace I've visited in the world. The present political problems are extraordinarily unlikely to affect you (considering where you will visit) in any way whatsoever.