If you want to have a constant level, try to use a compressor in your effect chain or apply a compressor effect with your recording software.
For your rhythm issue, there is no other "simple" solution than getting better at playing the song and play with a metronome.
You could correct the rhythm in post-production (editing your sound with a recording software), but this is a lot of work (but this works really well with drums) and would need some training to do it correctly (it's called quantization).
Please also keep in mind that if you are using some effect or some gear to record, there might be some latency involved (everything that you play would then be a few milliseconds late on the beat).
Your mic might also be catching some of the room's reverberation, which could mess up your sound.
A software I really like to use for recording and that has a lot of features is Reaper which is free to try and quite cheap. It needs a bit of work to master it but it's an awesome piece of software.
You still didn't tell us how you are recording. Your issue might also be a mic issue (it can be a shitty gear issue).
How are you miking ?
Is it a electro-acoustic guitar and you are recording directly from its output ?
Are you using a mic ?
Are you recording guitar and vocals at the same time (same mic, different mics...) ?
Are you using any effect on your guitar that might mess up a bit your recording (too much gain, improper setup of a delay, too much reverb...) ?
How are you keeping the tempo (metronome, click track, drum track...) ?
More important : from another point of view (get someone to listen to you "live"), do they think you are not playing in rhythm ? (while you are playing, you can't be a good judge).
If it is the case, I guess you need to work more on your rhythm technique. Gear/Software won't fix anything if the issue is your playing.
Based on your answers, you should :
- try to use a compressor effect on the vocal track, it will level the volume.
- Do some experiments on vocals recording : you should use some pop shield (there are tutorials on the Internet on how to build one cheap) and be really careful about your positioning. Maybe try to not be too close to the mic. But maybe your mic is just too cheap.
- For your rhythm issue, try to set up a basic drum rhythm (with Guitar Pro maybe, or a software drum machine), it would be much easier to follow than a simple metronome.
- Record your song in several takes. You don't need to record it in one shot. If you mess up one section, just record another take on that section : start recording a few bars before the section you want to rerecord (in order to start correctly on the beat), and use a proper editing software (again, Reaper is really good) that will allow you to easily work with different takes and mix them (trimming, moving, mixing...).
This where a click track or drum track could also be really useful : to keep track of where you are, and start correctly on the beat.