Of course it doesn't lose, but it's also a pretty useless move. 1.c3 e5 2.d4 only becomes a QGD and 1.c3 c5 2.d4 only becomes a Slav if black cooperates by taking and then playing ...d5. Why would he do that?
Your first line looks pretty reasonable (and boring) except when I see the diagram I think "what is that pawn on c3 doing there?"
Why do you want to play 1.c3? What do you expect from it?
1. c3 e5 2. c4!
According to the game explorer of master games here, c3 was played in 188 games. Of those the biggest reply is Nf6 with 58 games, followed by d5 in 36 games and e5 in 26 games.
In the 1. c3 e5 2. d4 line, there was only 1 game with e4 and it ended up a draw.
My sugggestion, if you want to play the opening for fun, just do it. Openings have similar goals and if you continue to follow opening principles you won't get into too much trouble just because of the opening choice.
As to your worries about your Knight, it will probably deploy to d2; from there the position will probably decide where it should go.
Your first line looks pretty reasonable (and boring) except when I see the diagram I think "what is that pawn on c3 doing there?"
c3 is often a useful move in that sort of London set up.
First thought that comes into my head is that if you play 1. c3 then you are almost certainly going to play d4 at some point (probably on move 2), so you may as well just play 1. d4.
As IM pfren wrote, c2-c3 is itself not bad. But you should rather include it in a system, and judge that system's performance. The reason for this is that c2-c3 can itself not influence the game in such a characteristic way, as some more active other first moves would do. You could play c2-c3 later, unless you want to give Black some initiative; but at the latter case you should be prepared to face different Black choices, and it is up to how well 1.c3 allows you to do that to justify it as playing at the first move.
Also, 1.c3 is not very offensive, and if Black plays defensively I believe he should be able to easily draw. I think 1.c3 could be justified in case you know Black will go for the win in a particular game. It may turn out into a Caro-Slav colors reversed formation with a bonus tempo.
1.c3 isn't fatal in any respect, and can be a real chameleon. You'll need a plan to handle 1...e5 though, or risk some double-e-pawn lines that are better for Black.
1. c3 e5 2. c4!
This looks good at first, but black has the clever reply 2...e4! Let's face it, 1.c3 is the worst idea since the Yugo....
2. c4 (after 1. c3 e5) is certainly OK. It is plain normal sicilian with reversed colours.
1.b3 is fun if you want to play something different :) look at games by Larsen for example
Thank You for answers... I will consider if usage of this opening is rational.
Roton, it is definitely rational. The ONLY thing you are losing is you are giving up your blessing of controlling the flow of the game with the first move. Moves like 1. d4, e4, etc take up more central control - they are more commanding. 1. c3? Not so much.
That being said, it is very transpositional.
You mentioned 1. c3 e5 2. d4 e4 being something you fear. Why not play 3. c4? That is a type of a reversed Caro-Kann. You can later attempt to put pressure on the e4 pawn with f3.
I think you should try it. You are interested in it - play it, learn something from it, and then either move on, or embrace it. Whatever you like - it is your chess.
What is the point of playing 1. c3 d5, 2. na3?
Na3? I think knight would be very badly placed as well. d2 is obvious square, but I prefer to develop knight on c3... I admit, that black would just help me with exchanging pawns in the center.
2. e4(?) is still quite annoying to me. I just feel squeezed in my position. My pieces can barely move at the very beginning.
Thank You all for suggestions! I found it very useful.
It's winning, of course!
It's winning, of course!
For black, possibly 
1 c3 is better than it looks.
If 1.....e5 then 2 d4 exd4 3 Qxd4 Nc6 then 4 Qa4 or Qd3
Now have a Scandinavian with an extra move.
I have done well as Black with the Qd6 Scandinavian and expect even better with this.
If 1....Nf6 then 2 d4 g6 3 Nf3 Bg7 4 Bf4 0-0 5 h3 d6 6 e3 Nbd7 7 Be2 Re1? 8 Bh2 e5?
9 dxe5 dxe5 10 Nxe5 winning a pawn (If Nxe5 Qxd8)
Note Black should play 7....Qe8 if he follows with e5.
Also 8 0-0 is not good because after winning the pawn you need to play Nd2 then Nf3 to activate your queens side.
This is the London opening against the Kings Indian set up and I have done OK with it.
If 1.....d5 then 2 d4 and Black has to be careful not to play Bf5 or Bg4 too soon because of Qb3
White can either get the queens bishop out then e3 or if an early c5 from Black can play e3 (A reversed Slav with extra move).
The main advantages of 1 c3 are:
1. Opponent will underestimate it and you.
2. Good in rapid games (make him/her think).
I am preparing a suprise opening for a small tournament (of quite weak players like me; around 1500). Does anybody have any experience with this opening? I have serious doubts due to knight development (c3 square is occupied).
Is this (short) variation with d5 correct?
What about c3 e5? I am really afraid of e4 after c3 e5 d4.
In c3 c5 d4 etc. becomes Slav?
Please, tell me some traps or interesting variations!