The primary resource for MEME-ChIP procedural questions is this paper:
Keep in mind that the running time of MEME grows as the square of the overall size of the sequence file, and as the cube of the number of sequences. ChIP-Seq data is usually highly redundant, with many nearly identical sequences from the same motifs. The random sampling of sequences in MEME-ChIP is to avoid wasting hours of computer time examining sequences that really have already been accounted for. If you are performing some sort of filtering that significantly reduces the redundancy of the ChIP-Seq data then yes, that might be a problem, but you'd have to filter out a LOT of data. The public MEME web site won't let you upload more than 60kb of sequence data. You can analyze larger files using a local copy of the MEME command line tool, but if you hand MEME more than a couple megabytes of sequence data, the running times will become completely impractical, even on a big cluster.
and maybe include also a number of other random peaks in order to achieve some statistical significance
There is no statistical benefit to including peak data you know to be unrelated to your regions of interest. In fact it may even reduce the statistical significance of your results! It can happen that the presence of a strong motif can prevent MEME from discovering a weaker motif, in which case MEME's discriminative mode might be useful. Check out the online help for discriminative mode on the MEME web form.
I am not sure if the requirements of MEME-ChIP -mainly same size sequences- are good for this study)?
ChIP-Seq will typically produce reads that are centered on the protein-binding region. This is why MEME-ChIP trims sequences to the central 100bp. You'll have to decide if this is appropriate for your experiment.
Finally, another question. Could a motif such as TTTTTTTTATTTTTTTT be valid? When using MEME I obtain several similar results, but... I am not quite sure about it, even it TOMTOM brings similar already known motives. Are they usual or an artifact?
Such a motif could certainly be valid as far a MEME and DREME are concerned, but keep in mind that MEME and DREME are only looking for short sequences that are statistically overrepresented. Neither MEME nor DREME can make any claims for biological significance. You'll have to use your judgement for that. Repeats and low complexity regions can generate as highly statistically significant motifs and may prevent MEME and DREME from finding motifs that are biologically more interesting, but have weaker statistical evidence. You may want to exclude known repeats or low complexity regions from your analysis. A motif like 'TTTTTTTTATTTTTTTT' may match surprisingly well to some known motifs, but in fact it will match pretty well to anything that has a run of 'T' somewhere in it.